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A 


JWi r 

l (p y (t, 

GEOMETRICAL TREATISE 

OF 

CONIC SECTIONS. 

IN FOUR ROOKS. 

TO WHICH IS ADDED, 

A TREATISE on the PRIMARY PROPERTIES 

OF 

V. 

CONCHOIDS, THE CISSOID, THE QUADRATUIX, 

CYCLOIDS, THE LOGARITHMIC CURVE, 

' AND THE 

LOGARITHMIC, ARCHIMEDEAN, AND HYPERBOLIC SPIRALS. 


BY THE 

Rev. ABRAM ROBERTSON, A.M. F.R.S. 

A *7— 

SAVILIAN PROFESSOR OF GEOMETRY IN THE 
UNIVERSITY OF OXFORD. 



OXFORD: 

AT THE CLARENDON PRESS. 


1802 


















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TO THE REVEREND 


CYRIL JACKSON, D. D. F. R. S. 

DEAN OF CHRIST CHURCH, 

I 

EQUALLY EMINENT FOR HIS OWN ABILITIES AND LEARNING, 

AND FOR 

HIS UNIFORM ENCOURAGEMENT AND PROMOTION 

OF TALENTS AND ACQUIREMENTS IN OTHERS, 

AS A 

l 

TESTIMONY OF THE HIGHEST ESTEEM; FOR HIS CHARACTER, 
AND AS A 

TRIBUTE OF GRATITUDE FOR MANY IMPORTANT FAVOURS, 

THIS WORK 

IS MOST RESPECTFULLY INSCRIBED, 

BY HIS MUCH OBLIGED AND FAITHFUL SERVANT, 


THE AUTHOR 






















• ' , 

, - 







■ 






\ * ■>' ' . / 
























ADVERTISEMENT. 


There are two points of which it feems necef 
fary that the reader of the following Treatife of 
Conic Sections fhould he apprized ; firfl, what pre* 
vious knowledge will he expended from him ; and fe- 
condly, what extent of information the Treatife itfelf 
is intended to afford. The firfl of thefe will prevent 
the young Jludent from entering upon the work till 
he is duly prepared, and the fecond will enable him 
to judge how far it is likely to contribute to the at- 
tainments which he has in view . 


It is expelled then, that the young Jludent fhould 
underjland thoroughly the firfl fix Books of Euclid\ 
the firfl twenty-one Propofitions of the eleventh 
Book, the two firfl of the twelfth, and the firfl prin¬ 
ciples of Algebra, and Pla?ie Trigonometry . 

As no number can be affigned as a limit to the 
properties of the conic fedlions, any treatife on the fub- 
jetl can be fiuppofed to contain only a feledlion of thofe 
which are mofl important and mojl ufeful, either 
generally, or with reference to the particular defign 
of the Writer. In the prefent infiance the defign 
has been, to furnifh the young Mathematician with 
b 2 fuch 




VIII 


ADVERTISEMENT. 


fuch a feries of prop ofi lions as might prepare him for 
confidering fome of the mojl important truths in fci- 
ence, and enable him to enter on the Jludy of 71a- 
tural philofophy, with the profped of obtaining a 
thorough knowledge of the fubjeff. According to 
thefe views the feledion of properties and the extent 
of the work have been regulated ; and at the fame 
time the arrangement and diviflon of the whole have 
been made with a deftgn of accommodating two de- 
fcriptions of readers. Thofe who are confidered as 
conflituting the firfi clafs are fuppofed to be defirous 
of a general but refpedable portion of knowledge of 
the fubjed . For the ufe of fuch a perufal of the 
jirjl three Books will be foundfuflflcient, as they con¬ 
tain the properties of the fedions mojl frequently re¬ 
ferred to in pure and mixed mathematics . For thofe 
who rank under thefecond, or higher defcription, a 
knowledge of all the four Books will be requiflte, as 
they complete the original deftgn of rendering the 
whole a preparative for the Newtonian Philofophy . 
The Author flatters himfelf indeed , that he flhall be 
found to have carried his elucidations of the Prin - 
cipia, in the prefent work, conflderably beyond what 
have been attempted in other treatifes of conic flec¬ 
tions. 

Something muft now be added concerning the par¬ 
ticular method, which has been adopted in thefe flieets, 
of deducing the primary properties of the J'etlmis 
from the nature of the cone. 


It 


advertisement. 

It is well known, that about the middle of the ft* 
venteenth century a difference of opinion took place 
among mathematicians concerning the proper fource 
from which the properties of the conic feCtions Jhould 
be deduced\ But notwithjlanding the objections which 
then began to be made to their deduction from the cone , 
and which have fince been continued, it appears to the 
Author of this work that the difficzdties attributed to 
the deductions from it were not to be imputed to the 
folid itfelf, but that they were occafionedfolely by the 
manner in which the deductions had been made 
The early writers did not happen to perceive that 
the general and extenfive property , exprejfed in the 
thirteenth Propofition of the firjl Book of this Trea- 
tife, could eafily be obtained from the cone ; and, 
not adverting to this, their deductions from the cone 
were fometimes tedious and intricate . 

The above-mentioned property, as far as fecants 
are concerned, occurs (I believe for ihefirjl time) in 
a folio volume, of which a treatife of co?iic feCiions 
makes a part, entitled, Euclides Adau&us et Me- 
thodicus, &c. publijhed by Guarinus in 1671. The 
property to the fame extent is to be found in Jozies's 
Synopfis Palmariorum Mathefeos, publijhed in 
1706; but neither of thefe two authors confidered 
the property as a fundamental one, ?ior do they feem 

* For foundations for fyftems, independent of the cone, fee 
the Scholium in page no, and the firft feVen articles in the 
Scholium at the end of the third Book. 


X 


ADVERTISEMENT. 




to have been aware of the advantages it was cap a* 
hie of producing . Its extenfive utility was firjl 
evinced in Hamilton s Conic Sections, publifhed in 
Latin in 1758 ; and on the appearance of this work 
ohjedlions to the cone ought to have ceafed. 

This was my perfuafwn when I publifhed my for¬ 
mer treatife and every deliberation on the fubjeCt 
fince has tended to Jlrengthen my conviction of its 
juftice for the following reafons. Firjl , the whole 
trouble with the cone is reduced to a very few de¬ 
monfir ations y for which no farther knowledge of 
Euclid is neceffary than what is requifite for Sphe¬ 
rical Trigonometry . Secondly , by this method the 
general properties are obtained with moji eafe and 
elegance . Lajlly, by deducing the properties from 
the cone the treatife is rendered more extenjively 
ufeful. No work on conic feCtions, confined to their 
defcription on a plane, can be applied to elucidations 
in PerfpeCtive, Projections of the Sphere, the Doc¬ 
trine of Ecligfes, and in fome other particulars of 
the higheji importance in fcience. 

For the refi it need only be faid, that the manner , 
in which the properties of the feCtions are clajfed 
and arranged, appeared to the Nut nory on the whole 9 

* In the year 1792 the author of the prefent work publirtied 
a quarto volume, entitled, SeBionum Comcarum Libri feptem. 
Accedit TraBatus de Seffiomkus Conicis , et de Scriptoribus qui 
earum doBtmam tradidcrunt. The laft mentioned Trad con¬ 
tains a full hiflorical account of the fubjed, 

to 


ADVERTISEMENT# 


xi 


to be that which was heft calculated to Jhew what 
properties are general, and what are appropriate to 
each of the feffions. 

The treatife following that on conicfe&ions, in the 
prefent volume, contains only the mofi common pro¬ 
perties of the curves fpecified in the title page. It 
is intended as a preparative for thofe who wijh to 
invejligate the higher properties by means of Flux¬ 
ions. In the firfi fettion methods of finding two mean 
proportionals and trifelling an angle, by means of 
conchoids, are inferted . In the thirdfeffion a method 
of dividing an angle in a?iy given proportion, by 
means of the quadratrix, is given ; as is alfo the 
quadrature of the circle, by means of the fame curve. 

N. B. When a demonfiration, in the following 
work, is effected by means of two ranks of magni¬ 
tudes, which, taken two and two in the fame or in 
a crofs order*, have the fame ratio to one another, 
they are placed thus, 

a : b : c : d 
E : p : G : h ; 

A, E, c, d reprefenting the firfi rank , and E, g, h 
the fecond. Previous to this arrangement of the 
7 ?iagnitudes, their ratio to one another is efiablijhed, 
and therefore it evidently appears in which of the 
two orders the magnitudes are proportional. In 

* That is either ex aeqnali, or ex aequali in proportione per- 
turbata. 

order 


ADVERTISEMENT. 


order to arrange the magnitudes in this manner . it 
was neceffary in two or three places to ufe this mode 


of contraction , which means the fquare of 



Xll 


a b if a tangent^ or the reft angle under its fegments 



/. 


ERRATA, 


Page 14. in the Corollary read can cut a fcalene cone 
— iz8. line 4. from the bottom, for it is a fecond diameter read c S' 
a fecond diameter 


— 226. line 9. read Nicomedes 

— 240. line 21. read Dinoftratus 

— 247. line *}.for b d read e d 

— 247. in the margin read Fig. 13. 

— 258. line 16, &c. read analogous to a fer^es of logarithms, and the 

ordinates bf,cg,dh, &c. are analogous to the natural 
numbers of thefe logarithms. 

■«— 259. line 15. and i%.f»r curve read fpiral 


xii 




LEMMAS 



LEMMAS 


. - \ 

FOR THE 

FIRST THREE FOLLOWING BOOKS 

OF 

CONIC SECTIONS . 

i 


LEMMA I. 

If the plane figure A F D be bounded by the Jlraight line Fig. t* 
A D and the curve a f d, and if the fquare . of the 
Jlraight line F I, drawn from any point F in the curve 
perpendicular to A D, be equal to the re&angle under the 
fegments A I, I D, the figure will be a femicircle . 

For let a d be bife&ed in c, and draw c F. Then. 

(47. i.) the fquare of c F is equal to the fquares of c 1, 

1 f together, and therefore, by hypothecs, equal to 
the fquare oPc 1 together with the rectangle under a i, 

1 d. But as a d is bife&ed in c, the fquare of a c or 
c D (5. ii.) is equal to the fquare of c 1 together with 
the re&angle under a i, i d. Confequently the fquare 
of c f is equal to the fquare of a c or c d ; and there¬ 
fore c f is equal to a c or c d. The figure a f d is 
therefore a femicircle. For Prop. IV. Book I. 


B 


LEMMA 





LEMMAS FOR THE 


% 

LEMMA IL 

If two Jlrnight lines be parallel , and a plane pafs through 
each of them , the common fc&ion of thefe planes, if they 
cut one another, will be parallel to each of the parallel 
lines . , 

fig. 2. Let two ftraight lilies as A d, B c be parallel. Let 
the plane a d f pafs through a d, and cut the plane 
B c Fj pafiing through b c, in the ftraight line E f ; 
the line of common fe&ion e f is parallel to A b, b c, 
For let thefe planes be cut by two parallel planes eab, 
f d c. Let the plane e a b cut the plane a b c d iu 
the ftraight line a b, the plane A d f e in a e, and 
the plane b c f e in b e. Let the plane f d c cut the 
plane A b c d in the ftraight line d c, the plane A d f E 
in d f, and the plane b c f e in c f. Then the ftraight 
line A b is parallel to d c (16. xi.) ae is parallel to 
d f, and b e is parallel to c f. Hence (34. i.) a b, 
d c are equal; the angle e a b (10. xi.) is equal to the 
angle fdc, and the angle e b a is equal to the angle 
f c d. Confequently (26. i.) a e, d f are equal, and 
therefore (33. i.) a d, e f are equal and parallel. For 
the fame real'ons e f, b c are parallel. For Prop. VII. 
Book I. 

LEMMA III. 

If a flraight line cut either of two parallel Jlraight lines, 
and be in the fame plane with them, it will, if fuffi- 
ciently produced, cut the other. 

Fig. 3. Let the ftraight lines a e, d e be parallel, and let 
c f be in the fame plane with them, and cut a b in the 
point c; the ftraight line c F produced will cut D E. 

For let any point g be taken in d e, and draw c g. 
Then as the angles b c g, e g c together (29. i.) are 
equal to two right angles, the angles f c g, e g c to¬ 
gether 


FIRST THREE FOLLOWING BOOKS. 


3 


gether are lefs than two right angles, and therefore 
(ax. 12. i.) the ftraight lines c f, d e produced will 
meet one another. For Prop. VIII. Book I. 

LEMMA IV. 

If two fraight lines cutting one another be parallel to a 
plane, a plane puffing through them will be parallel to 
the fame plane . 

Let the two ftraight lines a b, c b, cutting one an¬ 
other in b, be parallel to the plane d g h e ; the plane 
pafling through a b, c b is parallel to the plane 
DGHE. 

For let f be any point in the plane d g h e. Through 
A b and f let a plane be palled, and let it cut the plane 
r> g h e in the ftraight line d f h ; and let a plane pair¬ 
ing through c b and f cut it in e f g. Then will a b 
be parallel to dfh, and c e will be parallel to e f g. 
For if not, then a e will meet d f ii, and c b will 
meet e f g, and confequently ae,cb will meet the 
plane d g h e, in which d h, e g are, contrary to the 
hypothelis. The plane pafling through a b, c b (15.xi.) 
is therefore parallel to the plane d g h e. For Prop. 
X. Book I. 

LEMMA V. 

If the firjl of eight fraight lines be to the fecond as the 
third to the fourth, and if the fifth be to the fxth as the 
feventh to the eighth ; then the re It angle under the firjl 
and fifth will be to the rectangle under the fecond and 
fixth as the reltangle under the third and feventh to the 
rectangle under the fourth and eighth. And if the rect¬ 
angle under the firfi and fifth, of eight fraight lines, be 
to the rectangle under the fecond and fixth as the rc5l- 
angle under the third and feventh to the re Etangle under 
the fourth and eighth, and if the firf be to the fecond as 
E 2 the 


Fig. 4- 


4 


LEMMAS P6K tfHE 


the third to the fourth , then the fifth will he to the fixth 
as the feventh to the eighth . 

Fig. $. Part I. Let A B, the firft of eight ftraight lines, be 
to b c the fecond, as de the third to e f the fourth, 
and let g b the fifth be to b h the fixth as i e the fe¬ 
venth to e K the eighth ; then theredangleunder ab, 
g b is to the redangle under b c, b h as the redangle 
under d e, i e to the redangle under ef,ek, 

For let a b, b c be in a ftraight line; g b, b h be in 
a ftraight line; de, e f be in a ftraight line; and i E, 
E k be in a ftraight line ; and let thefe ftraight lines be 
at- right angles to one another, and let the redangles 
be completed as reprefented in the figure. Then a g is 
the redangle under a b, g b ; c h is the redangle un¬ 
der b c, b h ; d i is the redangle under de, ie; and 
F k is the redangle under e f, e k. By hypothefis 
A E : b c : : d e : e f, and therefore (n. v. and I* vi.) 
Ag:gc::di:i f. Again, by hypothefis, g b : 
B h : ; I E : E k, and therefore (n. v. and i. vi.) Q c ; 
c H : : I f : f k. Confequently, 

A G : G c : c H 
r> i : i F : f k, 

and therefore (22. v.) ag : ch : : di : fk, 

Fig. 5. Part II. The conftrudion, with refped to the red¬ 
angles, remaining as ftated above, let the redangle a g 
be to the redangle c h as the redangle d i to the red- 
angle f k, and let a b be to b c as d e to e f j then 
G B is t .0 B H as I E tO E K. 

For, by hypothefis and inverfion, bc:ab::bf: 
d e ; and therefore (11. v. and 1. vi.) g c : a g : : 1 F : 
d 1. Again by hypothefis a g : c 11 : : d 1: fk, Con¬ 
fequently, 

G C : A G : C H 
if: d 1 ;fk; 

and 


FIRST THREE FOLLOWING BOOKS. 


5 


and therefore ( 22 . v.) gc:ch::i f:fk. But (i.v i.) 
gc:ch::gb:bh, and if:fk::ie:er; and 
therefore (n,v.) gb:bh;:ie;ek, For Prop. 
X. Book I. 

LEMMA VI. 

If the points c, D be fo Jituated in the Jlraight line A B, 
that the rectangle DA c is equal to the retd angle cbd, 
then A c is equal to ED; or if the red angle a c b he. 
equal to the redangle B D A, then A c is equal to B D. 

Cafe I. Let CD be bife&ed in e, and then ( 6 . ii.) 
the re&angle dac together with the fquare of e c is 
equal to the fquare of ae; and the re&angle cbd to^ 
gether with the fquare of e d is equal to the fquare of 
B e. The fquares of ae,be are therefore equal, and 
confequently ae is equal to b e, and a c is equal to 
b d. 

Cafe II. Let a b be bife&ed in e, and then (5. ii.) 
the re&angle acb together with the fquare of ec is 
equal to the fquare of a e or e b ; and the re&angle 
b d a together with the fquare of e d is equal to the 
fquare of e b. The fquares of e c, e d are therefore 
equal, and confequently e c is equal to £ D, and a c is 
equal to b d. For Prop. I. Book II. 

LEMMA VII. 

If a Jlraight line touch a circle , and two Jlraight lines cut¬ 
ting the circle pafs through the point of contad, and meet 
a Jlraight line parallel to the tangent , the redangle 
under the fegments of the one, between the point of con- 
tad and circumference, and behueen the point of contad 
andftraight line parallel to the tangent , will be equal to 
the redangle under the fegments of the other , between the 
point of contad and circumference and between the point 
of contad and fraight line parallel to the tangent . 

* 3 


Fig. 6 . 


Let 


6 


LEMMAS FOR THE 


Fig. 7. Let the flraight line a b be parallel to the ftraight 
a g d line r g touching the circle epf in the point p, and 
let the two ftraight lines e p, f p, paffing through p, 
meet the circumference again, the one in e and the 
other in f, and let e p meet the ftraight line A b in c, 
and f p meet it in d ; then the rectangle under e p, 
p c is equal to the rectangle under fp, p d. 

For e f being drawn, the angle e f p (32. iii.) is 
equal to the angle rpe, which (29.1.) is equal to the 
angle p c d. The triangles e p f, d p c are therefore 
equiangular, and (4. vi.) ep:ff::dp:pc. Con- 
fequently, (16. vi.) the rectangle under e p, p c is equal 
to the redangle under f p, p d *. For Prop. I. Book 
III. 

Cor. 1. If a b cut the circle in b, and pb be drawn, 
it may be proved in the fame way that the redangle 
under f p, p d is equal to the fquare of pb. For the 
tangent r p being produced to g, and b f being drawn, 
the angle b p g (32. iii.) is equal to the angle b f p ; and 
(29. i.) it is alfo equal to the angle d b p. The trian¬ 
gles bfp, d b p are therefore equiangular, and (4. vi.) 
fp:pb::pb:pd, and (17. vi.) the redangle under 
f p, p d is equal to the fquare of pb, 

Fig. 8. Cor . 2. The reft remaining as above, if b g be drawn 
parallel to f p, b g is (34. i.) equal to p d, and there¬ 
fore by the above fp = 

B G 

LEMMA VIII. 

If the firfl of three flraight lines he to the third as the 
fquare of the fum of the fir ft and fecond to the fquare of 
the fum of the fecond and third, the fecond will be a 

* The flraight line a b may be on either fide of the tangent R c, 
and it is not neceflary, upon being produced indefinitely, that it fhould 
jrpeet the-circumference of the circle. * 

mean 


FIRST THREE FOLLOWING BOOKS.' 

mean proportional between thefrfl and third ; or 'if the 
frjl be to the third as the fquare of the difference of the 
frjl and fecond to the J'quare of the difference of the fe- 
cond and third , the fecond will be a mean proportional 
between the frjl and third , 

Part I. Let a denote the firft, B the fecond, and c 
the t hird, o f t he ftraig ht lines. Then by hypothecs A : 
c : : aT- 4 - b^ z : b 4 c v ~; and it is to be proved that B is 
a mean proportional between a and c. 

Let d be a mean proportional between a and c, 
and then, by inverfion, d : a : : c : d, and (18. v.) 
A + d : a : : d 4 c : d 5 and therefore by alternation 
a:d::a4d:d4c. Confequently (22. vi.) a 2 : 
d 2 : : a 4 D' z : d 4 c? 2 . But (Cor. 2. 20. vi.) A : c : : 
A 2 : d 2 , and therefore by hypothefis (and 11. v.) 
a 4 b 1 2 : b 4~c u : : a 4 d 12 : d 4 c' 2 . Confequently 
(22. vi.) A 4 B: B 4 c: : a 4 d: d 4 c; and there¬ 
fore, by converfion, a4b:a — c::a4 d : a — c, 
and (14. v.) A 4 b is equal to a 4 d. Confequently b 
is equal to d, and therefore b is a mean proportional be-, 
tween A and c. 

Part II. Let a denote the firft of the three ftraight 
lines, b the fecond, and c the third. Then by hypo¬ 
thefis a : c : : a — b 2 : !T^~c 2 $ and it is to be proved 
that b is a mean proportional between A and c. 

Let d be a mean proportional between a and c. 
Then a : d : : d : c, and by converfion a : a — d : : 
d : d — c; and by alternation, a: d : : a — d : d —c* 
Confequently (22. vi.) A 2 : d 2 : : a — d 2 : iT^c? 2 . 
But (Cor. 2. 20. vi.) a : c : : A 2 : D 2 ; and therefore by 
hypothefis (and 11. v.) A — b 2 : b — c' 2 : : aT— D Vi : 
d — c \ Confequently (22. vi.) A — B : b — c : ; 
A — d : d — c; and therefore (18. v.) a - c : b — c 
? : a — c : D — c. L[ence (14. v.) b — c is equal to 
14 P - C, 













8 LEMMAS, &GC* 

p — p, and therefore b is equal to D. Confequently b 
is a mean proportional between A and c. For Prop. 
XXIII. Book III. 


\j :rV< 



— 8 : " • 

ot iwpe is - 

.'J — u 


A GEO- 


A 


GEOMETRICAL TREATISE 

OF 

CONIC SECTIONS. 


BOOK I. 

Containing general Properties deduced from the Cone, 


DEFINITIONS. 

I. 

I F through the point v, without the plane of the Fig. 
circle a f b, a ftraight line avd be drawn, and ex¬ 
tended indefinitely both ways, and if the point v re¬ 
main fixed, and the ftraight line a v d be moved round 
the whole circumference of the circle, two Superficies 
will be generated by its motion, each of which is called 
a Conical Superficies ; and thefe mentioned together are 
called Oppofite Superficies . 

Cor. A ftraight line drawn from the fixed point v to 
any point G in either fuperficies is wholly in that fuper- 
ficies ; and, being produced, the part on the other fide 
of v is wholly in the oppofite fuperficies. For a ftraight 

line 







10 


GENERAL PROPERTIES 


BOOK 

I. 


' # 


line having the fame pofition which the generating line 
A v d had, when it pafted over the point g, is in the 
fuperficies; and, being produced, the part beyond v is 
in the oppofite fuperficies. Hence the Cor. is evident; 
for only one ftraight line can be drawn from v to G, as 
two ftraight lines cannot inclofe a fpace. 

II. 

The folid contained by the conical fuperficies and the 
circle a f b is called a Cone. 

III. 

The fixed point v is called the Vertex of the Cone . 

IV. 

The circle A f b is called the Bafe of the Cone . 

V. 

Any ftraight line drawn through the vertex of the 
cone to the circumference of the bafe is called a Side of 
the Cone . 

VI. 

A ftraight line v c, drawn through the vertex of the 
cone and the center of the bafe, is called th &Axis of the 
Cone . 

VII. 

If the axis of the cone be perpendicular to the bafe, it 
is called a Right Cone. 

VIII. 

If the axis of the cone be not perpendicular to the 
bafe, it is called a Scalene Cone . 

IX. 

A plane is faid to touch a conical fiperf cies, when it, 
meets the fuperficies, and when, being produced inde¬ 
finitely, in any direction, it falls without the fuper¬ 
ficies. 

X. 

A ftraight line which meets a conical fuperficies, and 
which, being produced both ways, falls without the 

fuper- 



DEDUCED FROM THE CONE, 


II 


fuperficies, is called a Tangent $ but a ftraight line BOOK 
which meets a conical fuperficies in two points, or each L 
of the oppofite fuperficies in one point, is called a Se- 
cant. 

XI. 

A ftraight line is faid to be parallel to a plane, when 
both being produced ever fo far, both ways, they do 
not meet. 

XIT. 

If a cone be cut by a plane, their common interfec* 
tion is called a Conic Sedion. 

XIII. 

The common interfecfion of any plane, not palling 
through the vertex of the cone, with the conical fuper¬ 
ficies, is called the Curve of a Conic Section. 

PROP. I. 

If a. cone be cut by a plane puffing through the vertex , the 
fedion ‘ivill be a triangle. 

Let the cone v a f b be cut by a plane palling through Fig. 9; 
v the vertex, and let v a b be the common interfe&ion 
of the cone and plane; the fedion v a b is a triangle. 

For let the plane, pafting through v, cut the plane of 
the bafe in the ftraight line (3. xi.) a b, and the cir¬ 
cumference of the bafe in the points a, b ; and let the 
ftraight lines v A, vb be drawn. Then, as the points 
v. A, b are in the plane cutting the cone, the ftraight 
lines v A, v b are wholly in the fame plane ; and as the 
points A, b are in the conical fuperficies, the ftraight 
lines v A, v b are alfo wholly in the fuperficies, by the 
corollary to the firft Definition. The ftraight lines v a, 
v b are therefore the common interfedions of the co¬ 
nical fuperficies and the plane cutting the cone; and 
confequently the fedion v a b is a triangle, 



GENERAL PROPERTIES 


Cor . If a plane, pafling through the vertex, cut a 
cone, it will cut the oppofite fuperficies in two ftraight 
lines, and only in thole two. For if the plane v a b be 
extended on both tides of the vertex, it will cut the op¬ 
pofite fuperficies in the ftraight lines v a, v e produced, 
and in them only. This is evident from the above, and 
the corollary to the firft Definition. 

PROP. II. 

If either of the oppofite conical fuperficies he cut by a plane 
parallel to the bafe of the cone , the common interjection 
of the fuperficies and the plane will be the circumference 
of a circle , and its center will be in the axis of the cone . 

Let the fuperficies v A b d, or its oppofite fuperficies,. 
be cut by a plane parallel to the bafe ab d of the cone, 
and let f g h be the common interfe&ion of the fuper¬ 
ficies and this plane ; f g h is the circumference of a 
circle, and its center is in v c, the axis of the cone, or 
in v c produced. 

Let c be the center of the bafe, and let the axis v c 
cut the plane fgh in the point i. From the point i, 
and in the plane f g H,draw any two ftraight lines I f, 
i g to the conical fuperficies. Through v i c, i f let a 
plane be pafled, and let it cut the fuperficies in the 
fide v f a, and the bafe of the cone in the ftraight line 
c a. Let a plane alfo be pafled through vie, ig, and 
let it cut the fuperficies in the fide vgb, and the bafe 
of the cone in the ftraight line c b. Then (i6. xi.) f i, 
a c, and alfo gi,bc are parallel to one another, each 
to each: and (29. i.) the triangles A c v, f 1 v, and 
alfo the triangles b c v, g i v are equiangular, each to 
each. Confequently (4. vi.) A c : f 1 : : v c : V I, and 
alfo vc:vi::bc:gi; and therefore (n. v.) A c : 
f 1 : : b c : g i, and as a c is equal to b c, f i is (14. v.) 
equal to g 1* In the fame manner it may be proved 

that 





V 

J 

C 

Via 



A B 


0 E 

K 


H 

c/ 



.7. Bn si re sc. 



















































DEDUCED FROM THE CONE. 

that any other ftraight line drawn in the plane f g h 
from the point i to the conical fuperficies is equal to 
F r ; and therefore fgh is the circumference of a cif*' 
cle, and the point I, in the axis y c, is its center. 

Cor . i. From this Proportion, and the firft Defini¬ 
tion, it appears that any circle parallel to the bafe of 
the cone, having its center in the axis, and its circum¬ 
ference in either of the oppofite fuperficies, may be 
taken for the bafe of the cone. 

Cor . 2. The folid contained by the conical fuperfi¬ 
cies v f g h, oppofite tovABD, and the circle fgh, 
is a cone. 


PROP. III. 

If a fcalene cone be cut through the axis by a plane per - 
pendicular to the bafe , of the Jides of the feiiton , meeting 
in the vertex , one will be the greatejl^ and the other the 
leaf of all the Jides of the cone. 

For let v n o p be a fcalene cone, of which v is 
the vertex, nop the bale, and c the centre of the 
bafe. Let the ftraight line v b be perpendicular to 
the plane of the bafe, and meet it in b. Draw the 
ftraight line b c, and let it meet the circumference of 
the bafe in the points p, n. Through the ftraight 
lines v b, e c let a plane be palled, and let it cut the 
cone; and let the fe6tion formed, with the cone, be 
the triangle v n p, as in the firft Propofition. Then 
as the plane of the triangle v n p pafles through c, 
it cuts the cone through the axis*; and, as it paffes 
through v b, it is alfo (18. xi.) perpendicular to the 


* As the reprefentation of the axis could not render the demonftra- 
tion more perfpicuous, it was intentionally omitted in the figures. The 
Reader will find the fame omifiion in the figure for Prop. IV. and V. 
and intentionally made^i’or the fame reafon. 


*3 

BOOK 

L 


Fig. ii. 
and 
22 . 


bafe 



H 


Gh NERALPROPERTIES 


BOOK bafc nop. Tf therefore the point b be farther from K 
than from p, it remains to be proved, that V n is greater 
. and v p lefs than any other fide of the cone. 

Let v o be any other fide of the cone, and let it 
meet the circumference of the bafe in o, and draw 
no. Then, as v b is perpendicular to the plane of the 
bafe, the angles v b n, v b p, v b o are right angles; 
and therefore {47. i.) the fquare of v n is equal to the 
fquares of v B, b n together; and the fquare of v o is 
equal to the fquares of v b, bo together; and the 
fquare of v v is equal to the fquares of v b, b p toge¬ 
ther. But (7. and 8. iii.) b n is greater than B o, 
and b o is greater than b p ; and therefore the fquare 
of b n is greater than the fquare of b o, and the fquare 
of B o is greater than the fquare of B P. Confequently 
the fquare of v n is greater than the fquare of v o, and 
the fquare of v o is greater than the fquare of v p. 
Of all the fides of the cone therefore, v n is the greateft, 
and v p is the lead. 

If the perpendicular v b fall into the circumference 
of the bafe, then B and p will coincide; and (referring 
to 15. iii. inftcad of 7. and 8. iii) the demonftration will 
be the fame ns above *. 

Cor . As there can be only one perpendicular to a 
plane (13. xi.) drawn from the fame point above the 
plane, it is evident from the demonftration of this Pro- 
pofition that only one plane can cut a cone through 
the axis, and be perpendicular to the bafe. 

* It ts evident that all the Tides of a right cone are equal to one 
another. 1'or» in this cafe, the perpendicular to the bafe,*drawn from 
v, will fall into c, the centre, according to the feventh definition. 


PROP. 


I 



deduced prom the cone. 


*5 


PROP. IV. 

■JLet the fcalene cone v n o p be cut by a plane paffng 
through the axis , and perpendicular to the bafe nop, 
and let the common fettion be the triangle v N p •, in the 
Jide v P take any point D, and in the plane of the tri¬ 
angle make the angle v D A equal to the angle v N p ; 
then if the cone be cut by a plane puffing through D A, 
and perpendicular to the triangle v N p, its common fec- 
tion A f D e with the cone will be a circle . 


BOOK 

I. 


Fig. 13. 


For let the fide v p be lefs than the fide v N, as in 
the preceding Propofition, and produce a d to t and 
n p to r. Then, as v n is greater than v p, the angle 
vpn (18. i.) is greater than the angle v n p. But the 
angles v n p, v d a are equal, by hypothefis ; and as 
the angle p d t is equal (15. i.) to the angle V d A, the 
angle vpn is greater than the angle p d t. The angles 
v p n, d p r together are therefore greater than the an¬ 
gles p d t, d p r together ; and confequently the an¬ 
gles p d t, dpr together are lefs than two right an¬ 
gles. If therefore the ftraight lines ad,np be fuffi- 
ciently produced they will meet. Let them be produced 
and meet in r ; and let the plane of the fe&ion afdb 
cut the plane of the bafe n o p in the ftraight line r s. 
In d a take any point 1, and let the cone be cut by a 
plane pafiing through 1 and parallel to the bafe; and 
let the fedlion formed be the circle h f k b, as in the 
fecond Propofition. Let the circle h f k b cut the tri¬ 
angle v n p in the ftraight line H 1 k, and the fe<$tion 
afdb in the ftraight line fib. Then as the fedlion 
h f k b is parallel to the plane of the bafe, and as thefe 
parallel planes are cut by the plane of the le&ion 
afdb, the common feclions (1 6. xi.) fib, s r are 
parallel; and as the plane of the fe&ion afdb, and 
the plane of the bafe n o p are perpendicular to the 

plane 



GENERAL PROPERTIES 


16 

BOOK plane of the triangle v n p, and cut one another in sr ; 
the ftraight line s R (19. xi.) is perpendicular to the 
plane of the triangle v n p. The ftraight line fib 
( 8. xi.) is therefore perpendicular to the plane v n p, 
and confequetitly (4. xi.) perpendicular to h k, a d. 
But, as the plane of the triangle v n p paftes through 
the axis, h k is a diameter of the circle h f k b by the 
fecond Proportion, and therefore (3. iii.) f b is bifeded 
in 1. Confequently (35. iii.) the redangle under k i, 
I h is equal to the fquare of f 1 or B 1. Again, as the 
circle h f k b is parallel to the plane of the bafe, and 
as thefe parallel planes are cut by the triangle vnp, 
the common fedions (16. xi.) h i k, n p are parallel. 
The angle a h i (29. i.) is therefore equal to the angle 
V N p, and confequently equal to the angle kdi. The 
angles (15. i.) aih, kid are alfo equal, and there¬ 
fore the triangles aih, kid are equiangular. Con¬ 
fequently (4. vi.) A 1 : 1 h : : k 1 : 1 d, and (16. vi.) 
the redangle under a i, i d is equal to the redangle 
under K 1, 1 h 5 and therefore, by the above, the red¬ 
angle under a 1, 1 d is equal to the fquare of f 1 or 1 b. 
The fedion afdb is therefore a circle, by the firft 
Lemma. 

The circle a f d b formed in a fcalene cone, in the 
manner mentioned in the Propofition, is called a Sub¬ 
contrary Scttlon, 

prop. y. 

If a conic fedion be a circle , and be not parallel to the bafe 
of the cone , it will be a fubcontrary fedion. 

Fig 13. Let the cone v n o p be cut by a plane not parallel 
to the bafe nop, and let the fedion afdb formed 
by it, with the cone, be a circle 3 a f d b is a fubcon- 
trary fedion. 

For let 1 be the point in which the axis of the cone 

meets 



DEDUCED PROM THE CONE. i; 

meets the circle afdb, and through i let a plane be book 
pafted parallel to the bafe, and let f k b h be the circle L 
formed by it with the cone, as in the fecond Propofi- 
tion. Let B f be the common fe£Uon of this circle 
with the circle afdb. Then by Prop. II. the point 
i is the center of the circle f k b h, and confequently 
b f is bife£ted in i. Through i draw in the circle 
Afdb the ftraight line ad at right angles to b f ; 
and through a d and v, the vertex, let a plane be 
pafted, and let v n p be the triangle formed by it with 
the cone, as in the firft Proportion. Let h k be the 
line of common fe£tion of the triangle vnp and the 
circle f k b h. Let l be any point in a d, and through 
l let a plane be pafted parallel to the bafe n o p, or to 
the circle fkbh. Let mceg be the circle formed 
by this plane with the cone, and let m e be its line of 
common feclion with the triangle vnp, and c l g its 
line of common fe&ion with the circle afdb. Then 
(16. xi.) the ftraight lines h k, m e are parallel, as are 
alfo bif,clg; and as a i b is a right angle, A l c is 
(29. i.) a right angle. Again, as the ftraight line a d 
bife&s the ftraight line bf at right angles, A d (Cor. 

1. iii.) is the diameter of the circle afdb. The 
ftraight line gc (3. iii.) is therefore bife£ted in l. But 
as 1 is the point in which the axis of the cone meets 
the circle a f d b, it is evident that the triangle vnp 
cuts the cone through the axis, and confequently by 
Prop. II. m e is a diameter of the circle mceg, and 
the point l is not its center. Hence the diameter m e 
(3. iii.) bife&s g c in l at right angles, and g l is at 
right angles to a d, m e, and therefore it is at right an¬ 
gles to the triangle (4. xi.) v n p. Confequently (18. 
xi.) each of the le&ions afdb, mgec is at right 
angles to the triangle vnp, and therefore as mgec 
is parallel to the bafe, the cone is cut by the plane 
C VNP 



GENERAL PROPERTIES 


28 

BOOK v n p pafling through the axis of the cone* and per-* 
L pendicular to the bale nop. Again as G l is at right 
angles to each of the two diameters m e, ad, the rect¬ 
angle under ml^le is equal to the redangle under 
dl,la 3 each of thefe reftangles (35. iii.) being equal 
to the fquare of g l ; and therefore (16. vi.) d l : l e : : 
ml : l A, and ( 6 . vi.) the angle ldeis equal to the 
angle a m l , or (29. i.) v n p. The circle afdb is 
therefore a fubcontrary feCtion. 

Cor. A conic fe&ion neither parallel to the bafe of 
the cone, nor a fubcontrary fe&ion, is not a circle. 

DEFINITIONS. 

XIV. 

Fig. 10. The cones v a b d, v f g ii, having the common 
vertex v, and whofe fuperficies are oppofite, being ge¬ 
nerated by the fauie line as in the firft Definition, are 
called Oppojite Cones. 

Cor. It is evident from this, and the fccond Propo¬ 
rtion, that if either of the oppofite cones be cut by a 
plane parallel to the bafe of either, the feClion will be a 
circle. 

XV. 

Fig. 15. If the plane vbe touch the conical fuperficies in the 
fide v b, and the cone v a b f be cut by the plane fdc 
parallel to the plane V b e, the feClion fdc, formed by 
the cutting plane and the cone, is called a Parabola. 

XVI. 

‘The plane vbe is called the Vertical Plane to the 
Parabola. 

Cor. i. As the cone may be indefinitely extended, it 
is evident that the parabola may alfo be indefinitely ex¬ 
tended; and as the parabola does not furround the 
cone, it is evident that its curve does not include a 
fpace, 

Cor. 



DEDUCED PROM THE CONE. 

Cor . 3. An indefinite number of ftraight lines parallel 
to v b may be drawn in the plane of the parabola. For 
the common fection of any plane palling through y b, 
and any point in the parabola, with the parabola (16. 
xi.) will be parallel to v b. 

XVII. 

If the cone VABcbe cut by a plane, and if the fee- 
tion d k l h, formed by the plane and the cone, fur- 
round the cone, and is not a circle, it is called an El- 
lipfd. 

XVIII. 

If the oppofite cones vabe, v m n, be cut by a 
plane vbe paffing through the vertex v, and if they 
be alfo cut by a plane parallel to v b e, forming with 
the oppofite cones the fe&ions fdc, a r s ; each of 
the fe£Iions fdc, q r s is called an Hyperbola , and 
when mentioned together they are called Oppofite Hy¬ 
perbolas .. 

XIX. 

The plane y e e is called the Vertical Plane to the 
Hyperbola, or Oppofite Hyperbolas. 

Cor . i. It is evident, as each of the oppofite cones 
may be indefinitely extended, that an hyperbola may 
be indefinitely extended ; and that its curve does not 
^include a fpace. 

Cor. 3. An indefinite number of ftraight lines pa¬ 
rallel to v b or v e may be drawn in the plane of the 
oppofite hyperbolas. For the common fe&ion of any 
plane palling through v b, or v e, and any point in ei¬ 
ther hyperbola, with the plane of the hyperbolas, will 
be parallel (16. xi.) to v b or v e. 

XX. 

A ftraight line in the plane of a conic fe&ion, which 
meets the curve, and which being produced both ways 
falls without it, is called a Tangent ; but a ftraight line 
c 3 which 


19 


BOOK 

I. 


Fig. 16. 


Fig. 17. 



20 


GENERAL PROPERTIES 


BOOK 

I* 


I-lg. 14. 


which meets the curve of a conic fe&ion in two points* 
or each of the oppofite hyperbolas in one* is called a 
Secant * 


SCHOLIUM. 

Although only the Parabola, Ellipfe, and Hyperbola* 
are denominated Conic Sections, the attentive reader 
will readily perceive from the foregoing Proportions 
and Definitions, that five different Sections may be 
formed by the interfe&ion of a cone and a plane va- 
rying its pofition, For if a ftraight line parallel to the 
bafe be within the cone and remain fixed, and a plane 
move about it as an axis, when the plane paffes through 
the vertex, the interfe&ion of the cone and plane will 
be a triangle, as in the firft Propofition. When the 
plane has moved from the vertex, but Hill cuts both 
the oppofite cones, the fe£tion formed in each will be 
an hyperbola, as in the eighteenth Definition. When 
the plane, proceeding in its motion round the fixed 
ftraight line, has arrived at a pofition parallel to that of 
a plane touching the cone in one of its fides, the fec- 
tion which it then forms with the cone is a parabola, 
as in the fifteenth Definition. In any other pofition of 
the moving plane, befides thofe already mentioned, an 
ellipfe or circle will be formed with the cone, ac¬ 
cording to the circumftances ftated in the feventeenth 
Definition, and in the fecond and fourth Propofitions. 

PROP. Vi. 

One Jlraight line , and one only , can be drawn to touch a 
conic febiton in a given point in the curve . 

Let g d h be a conic fe£tion, and let d be a given 
point in the curve; through d one ftraight line, and 
only one, can be drawn to touch the fe&ion. 


Let 



DEDUCED FROM THE CONE. 

Let v be the vertex of the -cone v 4 c b, and through 
d draw yea fide of the cone, meeting the bafe in the 
point c. Draw cf (17. iii.) touching the bafe, and’ 
through vc,cf let a plane pafs, and let its line of 
common fe&ion with the plane of the fe£iion g d h be 
D e. Then the ftraight line d e touches the fe&ion 
gdh, and no other ftraight line can touch it in d. 

For, as c f meets the bafe in the point c only, it is 
evident from the firft Definition, that eveiy ftraight 
line, excepting v c, drawn from v to the tangent c f 
will fall without the fuperficies vacb, The plane 
palling through v c, c f, therefore, can only meet the 
fuperficies in the ftraight line v c, and the curve of the 
fe&ion g d h in the point d only. Confequently as 
D e is in the plane palling through v c, c f,de touches 
the fe&ion gdh, according to the twentieth Defi¬ 
nition. 

But no other ftraight line can touch the fe£tion 
g d h in the point d. For, if it be pollible, let d i 
touch the fe&ion, and then as d 1 meets the curve 
gdh only in the point d, it can meet the fuperficies 
in that point only, and it will therefore touch the fu¬ 
perficies in d. Moreover as no ftraight line, except¬ 
ing d e the line of common fe&ion, can be in the plane 
of the fe&ion gdh and alfo in the plane vcf, and as 
D 1, according to hypothefis and the twentieth Defi¬ 
nition, is in the plane of the fe£tion g d h, d i is not in 
the plane vcf. Let a plane be palfed through the 
ftraight lines v d c, d i, and let it.cut the plane of the 
bafe in the ftraight line k c. Then as the ftraight line 
D 1 touches the fuperficies, every point in it, excepting 
D, falls without the fuperficies, according to the teiith 
Definition. It is therefore evident, from the firft De¬ 
finition, that every ftraight line drawn from v, except¬ 
ing v d c, in the plane palling through y d c, d i, will 
c 4 fall 


21 

BOOK 

I. 



GENERAL PROPERTIES 


BOOK fall without the fuperficies. The plane palling through 

_ vdc, d i will therefore meet the fuperficies in the 

flraight line vdc only; and confequently k c will 
touch the bafe acb. Again, as the planes v c k, y c f, 
cut one another in the ftraight line v c, the flraight 
lines kc, cf are not in the fame flraight line. The 
two flraight lines kc 3 cf therefore touch the circle 
A c b in the point c, which (16. iii.) is abfurd. Confe¬ 
quently no other ftraight line, befides d e, can be 
drawn to touch the fe&ion gdh in the point d. 

Cor. i. If a flraight line as c f touch the bafe of the 
cone in the point c, and from v, the vertex, the fide 
v c be drawn; a plane palling through c f, c v will 
touch the conical fuperficies in the fide v c ; and it is 
evident from the firfl Definition (and i. xi.) that this 
plane produced on the other fide of v will touch the 
oppofite fuperficies in c v produced. 

Cor. 2. If a flraight line as de touch the conical 
fuperficies, or a conic fedion g d h, and a fide y d c of 
the cone be drawn through d the point of contact, a 
plane paffing through this fide of the cone and the tan¬ 
gent d e will touch the fuperficies of the cone; and 
being produced beyond v, it will touch the oppofite 
fuperficies in v c produced. This is evident from the 
demonflration of the Propofition, and the preceding 
Cor. 

Cor. 3. If the fe£tion g d h be an hyperbola, the 
tangent d e cannot meet the oppofite hyperbola. For 
d e is the common .interfe&ion of the plane vcf and 
the plane of the fe&ion g d.h, and, by the firfl Cor. the 
plane v c F touches the oppofite fuperficies in c v pro¬ 
duced. It is therefore evident from the eighteenth 
Definition that the tangent d e cannot meet the oppo¬ 
fite hyperbola. 


PROP. 




./ B a sore sc. 



























































DEDUCED PROM THE CONE 


4 


PROP. VII. 

If a fir night line pafs through the vertex and fall without ’ 
the oppojite co?ies , two planes , and only two , can he 
drawn through it to touch the conical fuperfcies; and 
thefe planes will be on the oppoftc fides of a plane puff¬ 
ing through the fraight line , and cutting the hafe . 

Let the ftraight line y g pafs through v the vertex, 
and fall without the oppofite cones v a m b, dvej 
two planes, and only two, can be drawn through it to 
touch the fuperficies; and if a plane pafs through v g, 
and cut the bafe in the ftraight line c f, the tangent 
planes will be on the oppolite fides of the plane pafling 
through v g, c f. 

Firft let the ftraight lines v g, c f be parallel. Let 
c f be bife6ted in l, and draw i l m at right angles to 
c f, and let it meet the circumference of the circle in 
the points m, i. Let a plane pafs through the ftraight 
line v g and the point i, and this plane will touch the 
fuperficies. For let it cut the plane of the bafe in the 
ftraight line i k. Then as c f, v g are parallel, and as 
the plane of the bafe pafles through c f, and the plane 
v i k pafles through v g, the interfecftion i k of thefe 
planes will be parallel to c f, by the fecond Lemma. 
But as i m bife&s c f at right angles, it pafles through 
the center of the circle (Cor. i. iii.), and as i k, c f are 
parallel, and as the angle I l f is a right one, the an¬ 
gle k i l is alfo a right one (29. i.), and therefore 1 k 
(1 6. iii.) touches the circle in 1. Confequently, by 
Cor. 1. Prop. VI. the plane pafling through v 1, 1 k, 
or, as above, through v g, v 1, touches the fuperficies ; 
and in the fame manner it may be proved that the 
plane pafling through v G and the point m touches the 
fuperficies. 

Secondly, let the ftraight line v G be not parallel to 
c 4 cf, 


S3 

BOOK 

I. 


Fig. 18. 
and 
19. 


Fig. 18. 


Fig. 1 9 * 



24 


GENERAL PROPERTIES 


BOOK 

f. 


Pig. 16. 


c f, but let it meet it in k. Draw k i, k m (17. ill.) 
touching the bate in 1, m ; and then the plane palling 
through v Gy 1 Ky and alfo that palling through v G 
and m k, will touch the fuperficies, by Cor. 1. Prop. 

VL 

It is evident, in either cafe, that no other plane, be- 
lides the above-mentioned two, can pafs through v g, 
and touch the fuperficies; and that one of thefe tan¬ 
gent planes is on the one tide, and the other on the 
oppolite fide of the plane palling through v G, c f. 

PROP. VIII. 

If a conic fedion furround the cone, two Jlraight lines , and 
only two, parallel to one another, can be drawn to touch 
the fedion ; if the fedion does not furround the cone, no 
Jlraight line, parallel to a tangent, can be drawn to 
touch the J'edion ; but if the fedion be an hyperbola , one 
jlraight line, and one only , parallel to a taiigent, can be 
drawn to touch the oppojite hyperbola . 

Part I. Let the fe&ion d h l k furround the cone 
v A f b, and let the ftraight line G d touch the fe&ion 
in the point d : another ftraight line, and only one, 
.parallel to g d, can be drawn to touch the fe£tion. 

For let v d a be the lide of the cone palling through 
d, the point of contact, and let a plane pafs through 
v A, d g, and this plane will touch the conical fuper¬ 
ficies in the fide v a, by Cor. 2. Prop. vi. In this 
plane, and through v, the vertex of the cone, draw 
v t parallel to dg. Then v t will fall without the 
oppolite cones; and by Prop. VII. another plane can 
be palled through v t touching the conical fuperficies. 
Let this plane touch the fuperficies in the lide v l b, 
and let its interfe&ion with the plane of the fe&ion 
d h l k be li. Then as l 1 is in the plane touching 

the 



DEDUCED FROM THE CONE. 

the cone In the fide v l b, it meets the conical fuperfi- 
cies in the point l only. It will therefore meet the 
curve of the ie&ion d h l k in the point l only, and ’ 
confequently it will touch the fedlion : and as the plane 
tvli palfes through v t, and the plane of the fedlion 
pafles through d g parallel to v t, by the fecond Lem¬ 
ma l i is parallel to d g. And as no other plane pair¬ 
ing through v T can touch the conical fuperficies, be¬ 
sides the two t v A, T v b, it is evident, from the fe¬ 
cond Lemma, that no other ftraight line befides l i, 
parallel to d g, can be drawn to touch the fedlion 

D H L K. 

Part II. Let FDcbe.a fedlion which does not fur- 
round the cone, and let d g touch the fedlion in the 
point d. No other ftraight line, parallel to d g, can 
be drawn to touch the fedlion. 

For let v b e be the vertical plane to the parabola, 
or hyperbola, as in the fifteenth, fixteenth, eighteenth, 
and nineteenth Definitions. Let v d a be the fide of 
the cone palling through d, the point of contadt; and 
through v d a, d g let a plane pafs, and let it cut the 
vertical plane in the ftraight line vt. Then, by Cor. 2. 
Prop. VI. the plane v d g will touch the conical fu¬ 
perficies, and (16. xi.) d g, y t will be parallel; and 
as v t is in the plane, touching the conical fuperficies 
in the fide v d a, it will fall without the oppofite cones. 
Another plane, therefore, and only one, can be palled 
through v t to touch the conical fuperficies, by Prop. 
VII. But when the fedlion is a parabola, the other 
plane palling through v t and touching the fuperficies 
is the vertical plane v b e, which is parallel to the pa¬ 
rabola. When the fedlion is an hyperbola, then the 
vertical plane vee pafles through v t, and cuts the 
bale of the cone in the ftraight line b e ; and fuppof- 
ing t v l to be the other plane palling through v t, 

and 


BOOK 

I. 


Fig.15. 
and 

17. 



GENERAL PROPERTIES 


a6 

BOOK 


Fig. 17. 


and touching the conical fuperficies, the planes tvl, 
v d g are on oppofite Tides of v b e, by the feventh 
Propofition. Confequently the plane tvl cannot meet 
the hyperbola f d c. It therefore follows from the 
above, and the fecond Lemma, that if the fe6tion does 
not furround the cone, no ftraight line, parallel to a 
tangent, can be drawn to touch the fe&ion. 

Part III. Let f d c, q r s be oppofite hyperbolas, 
and let d g touch the hyperbola f d c in the point d. 
Then one ftraight line, and only one, parallel to dg 
can be drawn to touch the oppofite hyperbola q r s. 

For, every thing remaining as in the preceding part, 
through the fides v a, v l, in which the planes t v a, 
t v l, palling through t v parallel to d g, touch the 
fuperficies, let a plane be paired, cutting the vertical 
plane in the ftraight line v w and the plane of the hy¬ 
perbolas in the ftraight line d r. Then as v w, r d, 
l v are in the fame plane, and as (16. xi.) v w, R d 
are parallel, and l v meets v w, it will alfo meet r d, 
by the third Lemma. Let them meet in the point r. 
Then as the plane tvl touches the oppofite cone 
m v n in l v produced, it will meet the plane of the 
hyperbolas in the point r. Let the interleclion of 
thefe two planes therefore be r x ; and as the plane 
of the hyperbolas paffes through d g, and the plane 
t v l paftes through T v parallel to d g, by the fecond 
Lemma r x is parallel to d g ; and being in the plane 
touching the conical fuperficies, it will touch the hy¬ 
perbola o. r s in the point r. It is alfo evident, for 
the fame reafons as are mentioned above, that no other 
ftraight line parallel to r x, or d g, can be drawn to 
touch the hyperbola q r s. 

Cor. t . If a ftraight line touch a conic fe&ion, a 
ltiaight line drawn through any point within the fec- 
tion, and parallel to it, will meet the curve in two 

points. 



DEDUCED PROM THE CONE. 2 ] 

points. For let every thing remain as in the demon- book 
ftration of the Propofition, and let p be any point x * 
within the fe&ion. Through v t and the point p let Fi , r I5> x6 
a plane be palled, and let this plane cut the plane of and 
the fe&ion in the ftraight line h k ; and, by the fe- 
cond Lemma, h k will be parallel to g d the tangent, 
and alfo to v t. Now as the point p is within the 
fe&ion it is alfo within the cone, and therefore, by the 
hr ft Propofition, the plane palling through v t and the 
point p will cut the cone in twofides ; and asthefe two 
fides and v t, h k are in the fame plane, and v t, h k 
are parallel, h k will meet each of thefe two fides, by 
the third Lemma; and as h k is in the plane of the 
fe&ion, it muft meet the curve of the fe&ion in the 
fame points in which it meets thefe two fides of the 
cone. It is alfo evident from the Propofition, and the 
preceding part of this Corollary, that a ftraight line 
drawn through any point within the oppofite hyper¬ 
bola a r s, and parallel to g d, will meet the curve 
Q r s in two points. 

Cor. £. If a ftraight line meet the curve of a conic 
fe£tion in two points, two ftraight lines may be drawn 
parallel to it to touch the feciion, if it furround the 
cone; but if the feftion does not furround the cone, 
only one ftraight line parallel to a fecant can be drawn 
to touch the fe&ion ; and, if the fe&ion be an hyper¬ 
bola, only one ftraight line parallel to a fecant can be 
drawn to touch the oppofite hyperbola. For, let H, k 
be the points in which the fecant h k meets the curve 
of the fection ; and through h k and v, the vertex of 
the cone, let a plane be palfed, and, if the fe6tion fur¬ 
round the cone, draw v t in this plane parallel to H K. Fig. 16 . 
Then as this plane, by the Cor. to Prop. I. can only 
cut the oppofite fuperficies in ftraight lines drawn 
through v the vertex and the points h, k, it is evi¬ 
dent. 




2 $ 

book: 


15. 

and 

» 7 . 


GENERAL PROPERTIES 

dent that v t miift fall without the oppolite cones. 
Confequently, by Prop. VII. two planes can be paiTed 
through v t to touch the conical fuperficies, one on 
each fide of hkj and the interlections of thefe planes 
with the plane of the fe&ion will touch the fe&ion, 
and, by the fecond Lemma, thefe tangents will be pa¬ 
rallel to h k. If the fe&ion does not furround the 
cone, let the plane palling through the fecant h k and 
v cut the vertical plane v e e in the ftraight line v T, 
and, for the fame reafons as are mentioned above, v T 
will fall without the oppolite cones. Then through 
v t two planes may be palled, by Prop. VII. touching 
the conical fuperlicies. But, according to the demon- 
ftration of the Proportion, only one of thefe planes can 
meet the parabola, and one of them can meet the hy¬ 
perbola, and the other the oppolite hyperbola; and, by 
the fecond Lemma, g d, the interfe&ion of the plane 
t v a with the plane of the fe&ion f d c, will be pa¬ 
rallel to h k, and it x the interfeftion of the plane 
t v l with the plane of the oppolite hyperbolas, will 
alfo be parallel to fi k, and each of the ftraight lines 
g d, r x mull touch the fe&ion which it meets. 

PROP. IX. 

If a Jiraight line meet the curve of a conic feftion in two 
points, any Jiraight line parallel to it, drawn through a 
point within the fame febtion, or, if the ftdion he an hy¬ 
perbola, within the oppofite hyperbola, will alfo meet the 
curve oj the ftdion, in which it is drawn, in two points . 
And if a fraight line meet each of the curves of two op - 
pojite hyperbolas in one point, a Jiraight line parallel to 
it, drawn through any point in the playie of thefe fee- 
iions, will alfo meet each of the curves of thefe oppofite 
hyperbolas in one point . 


Part 



I’hitc III.page -iH . 



J.Basirt sc. 



























































\ - 

















- 




























































































































*• • 










Deduced f&om the cone. 


Part I. Let acdb be a conic fe&ion, and let the book 
flraight line c d meet the curve in the points c, d ; a L 
ftraight line, as ab, drawn parallel to c d through any Fig 
point p within the fe£tion will meet the curve in two 
points. 

For let v be the vertex of the cone v e g h f, in 
which the fe£tion is formed, and through c d and y let 
a plane be palfed, and let it cut a plane palling through 
a b and v in the ftraight line v t. Then the plane 
palling through c d and v mull cut the cone in the 
lides v c, v D, and, by the fecond Lemma, v t is pa¬ 
rallel to c d and alfo to ab. Again, as v t is in the 
plane v c d, and as this plane, by Cor. Prop. I. cuts the 
oppofite fuperficies only in y c, v d, or in thefe lines 
produced, it follows that v t falls without the oppolite 
cones. The plane palling through vt,ab will there¬ 
fore cut the cone in two lides, and, by the third Lem¬ 
ma, a b will meet each of thefe two lides in the fuper¬ 
ficies of the cone, and being in the plane of the lection, 
it mult meet the curve in the fame points. If the fec- 
tion be an hyperbola, it is evident, for the fame reafons, 
that a llraight line drawn parallel to c d, through any 
point within the oppolite hyperbola will meet the curve 
in two points. 

Part II. Let a c, b d be two oppofite hyperbolas, Fig. 21. 
and let the ftraight line c D meet the curve of one of 
them in the point c, and the curve of the other in the 
point D. Through the point p in the plane of thele 
fe£lions let the ftraight line ab be drawn parallel to 
cd; A B will meet each of the oppolite hyperbolas 
a c, b d in one point. 

For let v be the vertex of the oppolite cones, in 
which the hyperbolas are formed, and through c D 
and v let a plane be palfed, and let it cut a plane pall¬ 
ing through A b and v in the ftraight line v T. Then 

the 



30 

BOOK 

I. 


GENERAL PROPERTIES 

the plane patting through c d and v mull cut the op¬ 
pofite cones in the tides vc, vd; and, by the fecond 
Lemma, v t is parallel to c d and alfo to ab. Again * 
as v t is in the plane v c d, and as this plane, by Cor. 
Prop. I. cuts the oppofite fuperficies only in v c, v d, or 
in thefe lines produced, it follows that v t falls within 
the oppofite cones. The plane palling through v T, 
a e will therefore cut the oppofite cones in two fides; 
and from the above, and the third Lemma, a b will 
meet one of thefe two fides in the one fuperficies, and 
the other in the oppofite fuperficies. But as a b is in 
the plane of the oppofite hyperbolas Ac, b d, it mufi: 
meet the curve of one of the hyperbolas and the fuper¬ 
ficies, in which this hyperbola is, in the fame point. 
The ftraight line a b will therefore meet the curve of 
each of the oppofite hyperbolas in one point. 

Cor. If a ftraight line as c d meet the curve of a co¬ 
nic fe&ion in two points, it will fall wholly within the 
fe&ion, but being produced it will fall without the 
fe&ion. If a ftraight line meet each of the curves of 
oppofite hyperbolas in one point, it will fall wholly 
without the hyperbolas, but being produced it will fall 
on the one fide within one hyperbola, and on the other 
within the oppofite hyperbola. For it is evident from 
the demon fixation of the Propofition that cd in Fig. 20. 
is within the cone, and that being produced it mufi; fall 
without it; and in Fig. 21. it is evident that cd is 
without the oppofite cones, and that being produced it 
falls on the one fide within one cone, and on the other 
fide within the oppofite cone. 

PROP. X. 

If a ftraight line cut either or both of the oppofite conical 
fuperficies , and meet a ftraight line which is parallel to 
the bafe of the cone , and which cuts either of the oppofite 

Jupcr- 



DEDUCED FROM THE COKE. 


fuperficies y the redlangle under the fegments of the firjl BOOK 
mentioned line will be to the rediangle under the Jeg - ** 

ments of the other in the fame ratio , wherever the point 
of concourfe may be in the frjl mentioned line *. 

Let the ftraight line fgh cut either or both of the Fig. *6, 
eppolite fuperficies in the points g, h, and meet, in the * 3 ' 
point f, the ftraight line fde parallel to the bafe of 
the cone, and cutting either of the oppoftte fuperficies 
in the points e, d ; the re&angle under G f, f h will 
be to the rectangle under df,fe in the fame ratio, 
wherever the point of concourfe f may be in the 
ftraight line fgh. 

Cafe i. If the ftraight line f g h be alfo parallel to Fi^. 26 . 
the plane of the bafe, then the feClion g d h e, formed 
by the cone and the plane palling through fgh, 
fed will be a circle, by the fourth Lemma, and Prop. 

II. and therefore (35 or Cor. 36. in.) the rectangle un¬ 
der g f, f h will be to the rectangle under d f, f e in 
the ratio of equality. 

Cafe 2. Let fgh be not parallel to the bafe of the Fig. 27, 
cone. Through fed let a plane be palled parallel to 28, 
the bafe, and let the fe< 5 tion formed by it with the 
cone be the circle d e i k, as in the fecond Propofibon. 
Through the points g, h, and v, the vertex of the 
cone, let a plane be palled, and let it cut the fuperficies 
in the lides avi,evk, the plane of the bale in the 
ftraight line abl, and the plane of the circle d e i k 
in the ftraight line ifk; and in the plane v g h draw 
V l parallel to g h, and let it meet abl in the point l. 

* When two fecants meet one another, the fegments of either of the 
two are its parts between the point of concourfe and the points in 
which it meets the fuperficies ; and if a tangent meet a fecant, or an¬ 
other tangent, its magnitude is limited by the point of concourfe, and 
its point of contaft. 

v Then 



3 * 

BOOK 

l. 


GENERAL PROPERTIES 

Then as the ftraight lines a b l, i f k (id. xn) are 
parallel to one another, and the ftraight lines v l , 
f G H alfo parallel to one another, in the triangles 
V l b, h f k, the angles (29. i.) bvl,vbl in the one 
are equal to the angles k h f, h k f in the other, each 
to each; and in the triangles via,gfi, the angles 
v A l, A v l in the one are equal to the angles gif, 
1 g f in the other, each to each. The triangles v l b, 
h f k are therefore equiangular to one another, as are 
alfo the triangles v i a, g f i to one another. Hence 
(4. vi.) 

v l : l A : : G f : f I, and 
vl:lb::hf:fk, and therefore, by the 
fifth Lemma, v l 2 : a l xlb: :gfxfh:if X 
f k. But (35 or Cor. 36. iii.) if xfk = dfxfe* 
and confequently, 

vl 2 :alxlb::gfxfh :df X fe, 

Cor. 1. If the ftraight line fgh meet the ftraight 
line f p parallel to the bafe, and touching either l'uper- 
ficies in the point p, the rectangle under g f, f h will 
be to the fquare of f p in the fame ratio, wherever the 
point of concourfe may be in the line f g h, as is evi-» 
dent from the above (and 3 6 . iii.) And if the ftraight 
line m v, pafting through v the vertex, be parallel to 
the bafe, and meet g h in m, m v is to be confidered 
as a tangent; far as above, by fimilar triangles, 
v l : l a : : g m : m v, and 

v l : l b : : h m : m v, and therefore, by the 

fifth Lemma, v l 2 : a l x l b : : g m x m h : m v 

Cor. 2. If a ftraight line cut either or both the gp- 

pofite fuperficies, and meet a ftraight line parallel to 
the bafe, and touching or cutting either fuperficies; 
the rectangle under the legments of the firft mentioned 
line will be to the fquare of the tangent, or the rect¬ 
angle under the fegments of the fecant which it meets 

in 


1 



DEDUCED FROM THE CONE. 


33 

m a conftant ratio, wherever the point of concourfe BOOK 
may be in the firfi: mentioned line. For this ratio will L 
be either that of equality, as in the firfi: cafe of the de- 
mon fixation, or it will be that of v l 1 to al X lb, as 
in the fecond cafe, and in the preceding Cor. 

PROP. XT. 

If a Jlraight line touch either of the oppofte fuperficies , and 
meet a Jlraight line parallel to the baj'e of the cone , and 
which cuts either of the oppofte fuperfcies , the J'quare of 
the tangent will be to the rebiangle under the fegments 
of the fecant in the fame ratio , wherever the point of 
concourfe may be in the tangent . 

Let the fixaight line t f touch either of the oppofite 
fuperficies in the point t, and meet in the point f the 
ilraight line f e, parallel to the bafe of the cone, and 
cutting either of the oppofite fuperficies in the points 
g, e j the fquare of the tangent t f will be to the 
re&angle under g f, f e, in the fame ratio, wherever 
the point f may be in the tangent t f. 

Cafe i. If the tangent tf be alfo parallel to the 
plane of the bafe, then the fe&ion t g e, formed by 
the cone and the plane palling through tf, fe will 
be a circle, by the fourth Lemma, and Prop. II, and 
the fquare of t f (36. iii.) will be to the redangle g f, 
f e in the ratio of equality. 

Cafe 2. Let t f be not parallel to the bafe of the 
cone. Through f e let a plane be pafied parallel to 
the bafe, and let the fe&ion formed by it with the cone 
be the circle d g e, as in the fecond Propofition. 

Through the tangent t f and v, the vertex of the cone, 
let a plane be palled, and let it touch the fuperficies in 
the fide v t d, according to Cor. 2. Prop. VI. and let 
it cut the plane of the bafe in the ftraight line c l, and 
d the 


Fig. 22. 


Fig. 22. 


Fig. 23. 



34 

BOOK 

I. 


GENERAL PROPERTIES 

the plane of the circle dge in the ftraight line D f* 
In the plane vcl draw v l parallel to t f, and let it 
meet the bafe in the point l. Then as the ftraight lines 
CL, df (16. xi.) are parallel to one another, and the 
ftraight lines vl,tf alfo parallel to one another, in 
the triangles vcl, t d f, the angles (29. i.) cvl, 
v c l in the one are equal to the angles dtf,tdf 
in the other, each to each. 

The triangles vcl,tdf are therefore equiangular, 
and confequently (4. vi.) 

vl:lc::tf:fd, and by the fifth Lemma, 
v l 2 : lc 1 : : tf 1 : f d 2 . 

But (36. iii.) f d 2 is equal to the re&angle, under 
G f, f e, and confequently v l 2 : l c 2 : : tf 2 : G f X 

F E. 

Cor. 1. If a ftraight line, as t f, touch either of the 
oppofite fuperficies in t, and meet a ftraight line, as 
f d, parallel to the bafe of the cone, and which touches 
either fuperficies as in d ; the fquare of t f will be to 
the fquare of f d in the fame ratio, wherever the point 
of concourfe f may be in tf. If m v, palling through 
v the vertex, be parallel to the bafe and meet t f in 
m, then m v is to be confidered as a tangent; and as 
above v l 2 : l c 2 : : t m 2 : m v\ 

Cor. 2. If a ftraight line touch either of the oppofite 
fuperficies, and meet a ftraight line parallel to the bafe, 
and touching or cutting either fuperficies; the fquare 
of the firft mentioned tangent will be to the fquare of 
the tangent, or the reftangle under the fegments of 
the fecant which it meets in a conftant ratio, wherever 
the point of concourfe may be in the firft mentioned 
tangent. For this ratio will be that of equality, as in 
the firft cafe of the demonftration, or it will be that of 
v l 2 to l c 2 , as in the fecond cafe. 


PROP. 



DEDUCED FROM THE CONE. 

PROP. XII. 

If the JlrJi of two Jlraight lines he parallel to the fecond . 
and touch or cut either or cut both of the oppojite fuperf- 
cies, and if the fecond alfo touch or cut either , or cut 
both of the oppofte fuperficies, and if each of the two 
meet a Jlraight line parallel to the bafe of the cone and 
touching or cutting either fuperfcies; then the J'quare of 
the firfl, if a tangent , or the redangle under its feg- 
ments , if a fecant , will be to the fquare of the tangent 
or the redangle under the fegments of the fecant which 
it meets , as the fquare of the fecond , if a tangent , or the 
redangle under its fegments , if a fecant , to the fquare of 
the tangent , or the redangle under the J'egments of the 
fecant which it meets . 

Cafe i. If the firft and fecond ftraight lines, parallel 
to one another, be alfo parallel to the plane of the bafe, 
then the fquare of the firft, if a tangent, or the re&an- 
gle under its fegments, if a fecant, will be to the fquare 
of the tangent, or the re&angle under the fegments of 
the fecant which it meets in a ratio of equality, as in 
the firft cafe of the demonftration of Prop. X. and XI. 
And, for the fame reafons, the fquare of the fecond, if a 
tangent, or the re&angle under its fegments, if a fe¬ 
cant, will be to the fquare of the tangent, or the rect¬ 
angle under the fegments of the fecant which it meets 
in a ratio of equality. In this cafe therefore the Pro- 
pofition is evident. 

Cafe 2. Let the firft and fecond ftraight lines, pa¬ 
rallel to one another, be not parallel to the plane of the 
bafe. Suppofe a plane to pafs through each of the two 
parallel lines, and v the vertex, and then thefe planes 
will cut one another, and their line of common fe&ion 
will pafs through v, and, by the fecond Lemma, it will 
be parallel to each of the two ftraight lines parallel to 
d 2 one 


35 

BOOK 

I. 



36 General properties 

book one another. Let v l be their line of common fedion* 
as in Fig. 27. 28. and 23. and let it meet the bafe in 
the point l. Then the planes palling through the two 
parallel lines and v l, will cut the plane of the bafe in 
ftraight lines palling through l ; and each of thefe 
lines of common fedion with the plane of the bafe will 
cut or touch the bafe, as in the fecond cafe of the de- 
monftration of the tenth and eleventh Propolitions ; and 
the redangle under the fegments of either, if a fecant, 
or its fquare, if a tangent, will be (35 and 36. iii.) equal 
to the redangle under al,lb,la being a ftraight 
line cutting the bafe in the points b, a. Confequently 
by Cor. 2. Prop. X. and Cor. 2. Prop. XI. the fquare of 
the firft ftraight line, if a tangent, or the redangle un¬ 
der its fegments, if a fecant, will be to the fquare of 
the tangent, or the redangle under the fegments of the 
fecant, which it meets as the fquare of vl to the red¬ 
angle under al, lb. For the fame reafon, the fquare 
of the fecond, if a tangent, or the redangle under its 
fegments, if a fecant, will be to the fquare of the tan¬ 
gent, or the redangle under the fegments of the fe¬ 
cant, which it meets as the fquare of v l to the red¬ 
angle under al, lb. Hence (11. v.) if the firft of two 
ftraight lines be parallel, &c. 

SCHOLIUM. 

As every point in the curve of a Conic Sedion is alfo 
in the conical fuperficies, it is evident that all the Pro¬ 
portions demonltrated concerning ftraight lines touch¬ 
ing or cutting the conical fuperficies, or oppofite fuper- 
licies, may be transferred to ftraight lines, which in 
the fame manner touch or cut a conic fedion, or oppo¬ 
fite hyperbolas. 


PROP. 



DEDUCED PROM THE CONE. 

PROP. XIII. 

If there be four Jlraight lines in the plane of a conic fedion 3 • 
and if A B the firfl meet c B the fecond, and D E the 
third meet F E the fourth , and if the frjl be parallel to 
the third , and the fecond to the fourth , and if each of 
them either touch or cut a conic fedio 7 i 3 or cut oppofte 
hyperbolas ; then the fquare of A E, if a tangent, or the 
reft angle under its fegments , if a fecant , will be to the 
fquare of c B, if a tangent , or the red angle under its 
fegments , if a fecant, as the fquare of D E, if a tangent, 
or the redangle under its fegments, if a fecant, to the 
fquare of fe, if a tangent , or the redangle under its 
fegments , if a fecant . 

Cafe i. If the ftraight lines ab, c b 3 and confe- 
quently d e } f e 5 be each parallel to the bafe of the 
cone in which the fe&ion was formed, or the bafe of 
the oppofite cone, the fe&ion muft be a circle, as in the 
firft cafe of Prop. X. or Prop. XI, and the ratio above 
Rated will be that of equality. 

Cafe 2. Let a b, d e be not parallel to the bafe of 
the cone, in which, or in which and its oppofite, the 
fecRion or oppofite hyperbolas were formed; but let 
c b, F e be parallel to the bafe, and then the Propo- 
fition is evident from the twelfth Propofition. 

Cafe 3. Let neither a b nor c b, and confequently 
neither d k nor f e, be parallel to the bafe of the cone; 
but fuppofe b g, e h to be ftraight lines parallel to the 
bafe of the cone in which the fe&ion, or in which and 
the oppofite cone the oppofite hyperbolas were formed; 
and let b g, e h touch or cut either of the oppofite co¬ 
nical fuperficies. Then by the twelfth Propofition, the 
fquare of a b, if a tangent, or the re&angle under its 
fegments, if a fecant, will be to the fquare of b g, if a 
tangent, or the re&angle under its fegments, if a fe- 
d 3 cant, 


37 

BOOK. 

I. 

Fig. *4. 



_ 3 8 

BOOK 

L 


Fig. 29. 
3 °. 


GENERAL PROPERTIES 

cant, as the fquare of d e, if a tangent, or the rectan¬ 
gle under its fegments, if a fecant, to the fquare of 
e h, if a tangent, or the reCtangle under its fegments, 
if a fecant. Again, by the twelfth Propofition and in- 
verfion, the fquare of b g, if a tangent, or the reCtan- 
gle under its fegments, if a fecant, is to the fquare of 
c b, if a tangent, or the reCtangle under its fegments, if 
a fecant, as the fquare of e h, if a tangent, or the rect¬ 
angle under its fegments, if a fecant, to the fquare of 
f e, if a tangent, or the reCtangle under its fegments, 
if a fecant. Confequently, 

t. AB 2> | rt.BG 2 ^ rt.CB 2 

or > : < or > : < or 

f. A B r j h f. B G r j M. CB r 

t.DEq JLEHS Jt-FE 1 

or > : < or > : < or 

f. DE r J If. EH r J v f. F E r 
The fquare of a b therefore (22. v.) if a tangent, or 
the reCtangle under its fegments, if a fecant, is to the 
fquare of c b, if a tangent, or the reCtangle under its 
fegments, if a fecant, as the fquare of d e, if a tangent, 
or the reCtangle under its fegments, if a fecant, to the 
fquare of f e, if a tangent, or the reCtangle under its 
fegments, if a fecant. 

Cor. If A B, c b, d e, f e be tangents, then it is evi¬ 
dent (22. vi.) that ab:cb::de:fe. 

PROP. XIV. 

Any freight line parallel to the fide of a cone , provided it 
he not in the plane touching the cone in that fide , will 
meet one of the oppofite fuperfcies in one point , and in 
one point only . 

Let the ftraight line d c be parallel to v b a fide of 
the cone v a m b, but not fituated in the plane touch¬ 
ing 



DEDUCED FROM THE CONE. 

mg the cone in the fide v b ; the ftraight line d c will 
meet one of the oppofite fupcrficies in one point, and 
in one point only. 

Let a plane pafs through the parallels d c, v b, and 
as by hypothefis d c is not in the plane touching the 
cone in the fide v b, the plane palling through dc,vb 
mull cut the cone. Let it cut the oppofite fuperficies 
therefore in the ftraight lines a v, b v. Then as a y 
meets b v in v the vertex, and as it is in the fame 
plane with the parallels v b, d c, by the third Lemma 
a v, or a v produced, muft alfo meet d c. Let them 
meet in d. Then d c muft meet one of the fuperficies 
in d, and as it is parallel to v b, it is evident it cannot 
meet the other fuperficies. It is alfo evident, that it 
can meet one of the oppofite fuperficies in one point 
only; for on one fide of d it is entirely within the fu¬ 
perficies, and on the other entirely without it. 

PROP. XV. 

If a Jlraight line parallel to a fide of the cone cut either of 
the oppofite conical fuperficies , arid meet two Jlraight 
lines parallel to the hafe of the cone , and which cut ei¬ 
ther Juperficies) the fegments of the firfi mentioned line , 
between the fuperficies and the points of concourfe , will 
he to one another as the rectangles under the fegments of 
the J'ecants which it meets . 

Let the ftraight line d c, parallel to vb a fide of the 
cone v A m b, cut either of the oppofite fuperficies in 
the point d, and meet in the points e, r the ftraight 
lines e i f, l r t, which are parallel to the bafe of the 
cone, and cut either fuperficies in the points i, f, and 
Ly t ; then the fegment d e is to the fegment dr as 
the re&angle under ei, e f to the rectangle ^nder 

L R, R T. 

d 4 For 


39 


BOOK 

I. 


Fig. 29 . 
30. 



40 


GENERAL PROPERTIES 


BOO 

I. 


For through the parallels d c, v b let a plane pafs, 
and let it cut the plane of the bafe in the ftraight line 
ACB, and the fuperficies in A v, b v. Through each 
of the ftraight lines l r t, e i F let a plane pafs pa¬ 
rallel to the bafe a m b, and let o l n t, i f h g be the 
circles formed, as in the fecond Propofition. Let e g h, 
o r n be the interfe&ions of thele circles and the plane 
pafting through a v, b v. Let e g h meet a y in g 
and b v in h j and let o r n meet Avin o and b v in 
n. Then (16. xi.) the ftraight lines e g h, o r n, 
a c b are parallel, and therefore (34. i.) e h, c b, r n 
are equal to one another; the angle d ge (29.1.) is 
equal to the angle dac, and the angle deg to the 
angle dca. Hence (4. vi.) d e : e g : : d c : A c, 
and dr :or::dc : Ac; and therefore (n. v.) 
de:eg::dr:or. and (i6.v.)de:dr::eg: 
o r. Confequently (1. vi.) de:dr::egxeh: 

O R X R N. 

But (35. and 36. iii.) egxeh = eixef, and 
orx rn = lrxrt; and therefore d e : d r : : 
ei xefjlrxrt. 

Cor. If a ftraight line parallel to a ftde of the cone 
cut either of the oppofite c~»nical fuperficies, and meet 
two ftraight lines parallel to the bafe, and which meet 
either fuperficies ; its fegment between the fuperficies, 
and the firft of the two parallel to the bafe, will be to 
its fegment between the fuperficies and the fecond, as 
the fquare of the firft, if a tangent, or the re&angle 
under its fegments, if a fecant, to the fquare of the fe¬ 
cond, if a tangent, or the re&angle under its fegments, 
if a fecant. For every thing remaining as above, if 
e p be parallel to the bafe, and touch either fuperficies 
in p, e p will be in the plane of the circle ifhg and 
(36. iii.) the fquare of e p will be equal to the rectan¬ 
gle under ei, e f. And, for the fame reafons, if the 

point 




J Ila-sirt .»«• 








































































DEDUCED FROM THE CONE. 


4 * 


point r were without the circle, the fquare of a ftraight 
line parallel to the bafe, drawn from r and touching 
either fuperficies would be equal to the rectangle under 
L r, r t. If a ftraight line v k parallel to the bafe 
£>afs through v, the vertex, and meet c d in k, v k is 
to be confidered as a tangent. For d k : k v : : d c : 
c A, and dr:or::dc:ca. Hence (i i. v.) d k : 
K v : : d r : o r, and (16. v.) d k : D r : : k v : o r. 
But k v, r n being parallel to the bafe are parallel to 
one another, and therefore (34. i.) k v, r n are equal, 
and confequently (1. vi.) dk:dr::kv 2 :or x 

1 \ N. 


PROP. XVI. 

If a Jlraight line cutting a parabola or hyperbola be pa¬ 
rallel to a fide of the cone in which the JeEiion is formed ’, 
and if it meet two Jlraight lines which are parallel to 
one another , and meet the fame fe&ion , or the oppofite 
hyperbolas ; its fegment between the curve and the firjl 
of the two parallels will be to its fegment between the 
curve and the fecond , as the fquare of the firfi , if a tan¬ 
gent , or the reElangle under its fegments , if a fecant, to 
the fquare of the fecond , if a tangent , or the re El angle 
under its fegments , if a fecant . 

Suppofe the ftraight line b c to cut the curve of a 
parabola or hyperbola in the point a, and to be parallel 
to a ftde of the cone in which the fe£tion is formed, 
and let it meet the ftraight lines b d, c e which are 
parallel to one another, and meet the fame fe&ion, or 
the oppofite hyperbolas; then a b is to a c as the 
fquare of b d, if a tangent, or the rectangle under its 
fegments, if a fecant, to the fquare of c e, if a tangent, 
or the re&angle under its fegments, if a fecant. 

For, as b c cuts the curve of the parabola or hyper¬ 
bola 


book; 

1. 


Fig. 25. 


GINARAL PROPERTIES 


4» 

HOOK bolft in the point a, it will cut the conical fupcrficics 
'• in which the curve is formed in the fame point; and 
the (Might lines n n, c k will meet the conical fuper- 
licicH in the fame points in which they meet the curve 
of the fed ion, or the curves of the oppofite hyperbolas. 
If therefore n i>, c k bo parallel to the bafe of the cone, 
the Propofition is evident from the Cor. to Prop. XV. 
but if they tiro not parallel to the bafe of the cone, let 
n f, c o la', parallel to the bale of the cone, and let 
them meet the fame or the oppofite conical fuperficies. 
Then by the Cor. to Prop. XV. ah is to a c as tho 
fquare of n r, if a tangent, or the re&anglc under its 
fegments, if a fecant, to the fquarc of c o, if a tan¬ 
gent, or the rectangle under its fegments, if a fccant. 
Again, by Prop. Ml. (and 16. v.) the fquare of n f, if 
n tangent, or the rectangle under its fegments, if a fe- 
cant, is to the lquarc of c o, if a tangent, or the rect¬ 
angle under its fegments, if a focuut, as the fquare of 
ii d, if a tangent, or the reCt angle under its fegments, 
if ft fecant, to tho fquare of r r:, if a tangent, or the 
rectangle under its fegments, if a fecant. Confequcntly 
(t t, v.) a a is to a e as the fquare of B n, if a tangent, 
*n the rectangle under its fegments, if a fecant, to tho 
Iquure of e k, if a tangent, or the rectangle under its 
fegments, if a fecant, 

PROP. XVIr. 

// tun jiraight fines meeting ntn' another touch a conic fee - 
or oppofite hyperbolas, anil if a fecant parallel to 
one of them meet the other ami the jh'(light line joining 
the points of contact, the rcStangte under the fegments of 
the fecant between the curve and the tangent will be 
equal to the fquare of its fegment bet ween the tangent 
and the line joining the points of contact. 




J flasirc sc 






































































DEDUCED PROM THE CONE, 43 

Let the two ftraight lines a p, c f, meeting one ano- book 
ther in f, touch a conic fe&ion or oppofite hyperbolas 
in the points a, c, and let the ftraight line d p, parallel Fig 3I> r 
to c f, cut the fecftion or either of the oppofite hyper- 33> 34* 
bolas in d, p, and meet the tangent a f in g, and the 
ftraight line a c, joining the points of contact, in b; 
then the re&angle under d g, g p is equal to the 
fquare of gb. 

For by Prop. XIII. dg X gp:ag 2 ::cf 2 :af\ 

But as c f, b g are parallel, by Lemma V. (and 4. vi.) 
cp 1 ': af 1 : : gb 2 : A g 2 , and therefore (n. v.) d g x 
gp : ag 1 : : qb 1 : AG 1 , Confequently (14. v.) d g 

X G P 55 G B 2 . 

Cor. 1. The reft remaining as above, if from any 
point e in the tangent a f, there be drawn the ftraight 
line e h parallel to the tangent c f, and meeting a c 
in h, and if from the fame point e there be drawn any 
ftraight line E i l, cutting the fe&ion or oppofite hy¬ 
perbolas in 1 and l ; then the re&angle under 1 e, e l 
and the fquare of eh will be to one another as the 
fquares of the tangents, or the re&angles under the 
fegments of the fecants meeting one another and pa¬ 
rallel to il, eh. For from the point g draw g n pa¬ 
rallel to 1 l, and let it cut the curve or curves in m, 
and n. Then by Prop. XIII. iexel:mgxgn:: 
a e 2 : a g 2 ; and by fimilar triangles, and this Propo- 
fition, A e 2 ; ag 2 : : e h 2 : g e 2 or its equal d g x g p. 

Hence (11. v.) 1 e X e l : mg x g n : : e h 2 : d g X 
G p, and, by alternation, iexelieh 2 :: mg X 
g n : d G X G p. Confequently, by Prop. XIII. (and 
II. v.) the Cor. is evident. 

Cor. 2. If a ftraight line b h cut a conic feblion, or Fig. 35. 
oppofite hyperbolas in d, g, and meet in the points b, 3<5 * 
h two ftraight lines ab,ch which touch the fe&ion 
or oppofite hyperbolas in a, and c, and if b h meet 



44 

BOOK 

I. 


GENERAL PROPERTIES, &C. 

A c joining the points of contaft in e ; the re&angle 
under db,bg will be to the re&angle under g h, h d 
as the fquare of b e to the fquare of h e. For if the 
tangents a b, c h be parallel, the triangles (29. i.) 
a b e, c h e will be equiangular, and therefore, by the 
fifth Lemma, (and 4. vi.) in this cafe a b 2 : c h 2 : : 
B e 2 : h e 2 ; and by Prop. XIII. a b 2 : c h 2 : : D B X 
b G : g h x h d. In this cafe therefore (11. v.) the 
Cor. is evident. But if the tangents be not parallel, 
through h draw the ftraight line l k parallel to a b, 
and let it meet a c in k, and the curve or curves in the 
points l, m. Then the triangles a b e, k h e are (29. 
i.) equiangular, and as above ab 2 : h k 2 : : b e 2 : « e 2 . 
But by this Propofition h k 2 is equal to l h x h m, 
and therefore ab 2 :lh X h m : : b e 2 : h e 2 ; and by 
Prop.XIII. ab 2 :lh xhm::dbXbg:gh xhd. 
Confequently (11, v.Jdbxbgigh X h d : : b e 2 : 
«E 2 . 


♦ 


A GEO- 




Hale V~/>qgt 


J.Bitsirt m. 




































































A 


GEOMETRICAL TREATISE 

OF 

CONIC SECTIONS . 


BOOK II. 

Of the FJlipfe and Hj’perhola . 


DEFINITIONS. 

I. 

r V 

JL HAT point within an ellipfe or between oppofite 
hyperbolas, in which every flraight line, palling through 
it and terminated by the curve or oppofite curves, is 
bife£ted, is called the Center of the ellipfe, or the Center 
of the hyperbola or oppofite hyperbolas. 

II. 

Any ftraight line pafling through the center of an 
ellipfe, and terminated by the curve, is called a Dia¬ 
meter of the ellipfe. 

III. 

A ftraight line palling through the center of oppofite 
■hyperbolas, and terminated by the oppofite curves, is 

called 







46 

BOOK 

II. 


OF THE ELLIPSE AND HYPERBOLA* 

called a Tranfverfe Diamiter of the oppolite hyperbolas* 
or of either of the oppofite hyperbolas. And a ftraight 
line palling through the center of oppofite hyperbolas, 
and bife&ing a ftraight line not palling through the 
center, and terminated by the oppofite curves, is called 
a Second Diameter of the oppolite hyperbolas, or of ei¬ 
ther of the oppolite hyperbolas. 

IV. 

Any ftraight line not palling through the center of 
an ellipfe, or oppofite hyperbolas, terminated by the 
curve of the ellipfe or either hyperbola, or by the op¬ 
polite curves, and bife&ed by a diameter, is called a 
Double Ordinate to the bife6ting diameter \ and its half 
is limply called an Ordinate to it. 

V. 

The points in which any diameter of an ellipfe meets 
the curve, or in which any tranfverfe diameter of oppo¬ 
fite hyperbolas meets the oppolite curves, are called 
the Vertices of the diameter ; and the legments of a.dia- 
meter, between an ordinate and its vertices, are called 
Abfciffes. 

VI. 

Two diameters of an ellipfe, or oppolite hyperbolas, 
of which each bifedls all ftraight lines terminated by 
the curve, or oppofite curves, and parallel to the other, 
are called Conjugate Diameters . 

VII. 

A diameter of an ellipfe, or oppofite hyperbolas, 
which cuts its ordinates at right angles is called an 
Axis of the ellipfe, hyperbola, or oppolite hyperbolas. 

PROP. I. 

If two parallel ftraight lines touch an ellipfe , or oppo/ite 

hyperbolas , the flraight line joining the points of contain 

will 



OP THE ELLIPSE AND HYPERBOLA. 


.. 47 


"Will bifeEl any ftraight line parallel to them , and termi¬ 
nated by the curve of the ellipfe y or by the curve of either 
of the oppojite hyperbolas . 

Let the two parallel ftraight lines g h, l m touch 
the ellipfe a d b e, or the oppofite hyperbolas fbn, 
iAKj in the points b, a ; the ftraight line a b, joining 
the points of contact, will bifect any ftraight line f n, 
parallel to the tangents, and terminated by. the curve 
of the ellipfe, or by the curve of either of the oppofite 
hyperbolas. 

Let f n meet the curve in the points f, n, and a b 
in o. Through the points f, n draw g l, h m parallel 
to ab. Let g l meet the curve of the ellipfe again, 
or the curve of the oppofite hyberbola in i, the tan¬ 
gent g h in g, and the tangent l m in l. Let fi m 
meet the curve of the ellipfe again, or the curve of the 
oppofite hyperbola in k, the tangent g h in h, and the 
tangent lm in m. Then, by Prop. XIII. Book I. 
B G 2 : i g X g f ; : a l 2 : f l xlij and (34. i.) as 
b g is equal to a l, and therefore the fquare of b g 
equal to the fquare of a l, we have (14. v.) 1 g x g f 
equal to fl X l i. Conlequently, by the fixth Lem¬ 
ma, F G is equal to 1 l. For the fame reafons n h is 
equal to k m, and (34. i.) as f g is equal to N h, i l is 
equal to KM) and (34. i.) as l g is equal to m h, i g 
is equal to k h, and 1 g x g f is equal to K h x h n. 
Again, by Prop. XIII. Book I. 1 g X g f : g b* : : 
K h X h n : H B 2 ; and therefore (14. v.) the fquare of 
g b is equal to the fquare of h b, and g b is equal to 
H B. Conlequently (34. i.) f o is equal to on. 

If, in the ellipfe, the ftraight line d e, meeting the 
curve in d, e, be parallel to the tangents g h, l m, and 
if the ftraight line pdt parallel to a b touch the el¬ 
lipfe in d, then the ftraight line r e q parallel to ab 

will 


BOOK 

IL 


37 - 

3 *. 



48 

BOOK 

II. 


OF THE ELLIPSE AND HYPERBOLA# 

will touch the ellipfe in e, and d e will be bife&ed in 
c, the point in which it meets a b. For let p d t meet 
the tangent g h in t 5 and the tangent l m in p ; and 
let r e q meet the tangent g h in a, and the tangent 
l m in r. Then, by Prop. XIII. Book I. b t 2 : d t 2 : : 
A v 2 : pd 1 ; and (34. i.) as bt, a p are equal; the 
fquare of b t is therefore equal to the fquare of A p, 
and confequently (14. v.) the fquare of d t is equal to 
the fquare of d p, and d t is equal to d p. But (34. i.) 
e q, d t are equal to one another, as are alfo e r, d p 
to one another; and therefore e a, e r are equal to 
one another. Now if re q could meet the curve in 
any other point befides e, as in y, then as b a (34. i.) 
is equal to a n, it might be proved as above, by means 
of Prop. XIII. Book I. that v a x a e is equal to e r 
x r v. It would therefore follow, by the fixth Lem¬ 
ma, that e q is equal to r v ; which by the above is 
abfurd. Confequently r e cl touches the ellipfe in 
e, and therefore, by the Cor. to Prop. XIII. Book I. 
e q : a e : : d t : t b ; and as e q, d t are equal, we 
have (14. v.) b a equal to b t. Confequently c d is 
equal to c e, for (34. i.) c E is equal to b q, and c D 

to B T. 

Cor. If two parallel ftraight lines, as g h, l m touch¬ 
ing an ellipfe or oppofite hyperbolas meet a ftraight 
line, as f 1, which cuts the ellipfe or oppofite hyper¬ 
bolas, and is parallel to the ftraight line joining the 
points of contaft, the fegments of the fecant between 
the curve or curves and the tangents will be equal to 
one another ; for by the above f g is equal ton. And 
it two parallel ftraight lines touching an ellipfe meet a 
ftraight line which touches the ellipfe, and is parallel 
to the ftraight line joining the points of conta6I, the 
fegments of the laft mentioned tangent, between the 
point of contact and the parallels, will be equal to one 

another. 



OF THE ELLIPSE AND HYPERBOLA. 

another. This is alfo evident from the above, for it 
was proved that d p is equal to dt. 

PROP. II. 

If two parallel Jlraight lines touch an ellipfe or oppojite 
hyperbolas , and the Jlraight line joining the points of con¬ 
tact be bifebled , the point in which it is bfeEled will be 
the center of the ellipfe or oppofte hyperbolas; and no 
other point can be a center of the ellipfe or oppofte hy¬ 
perbolas . 

Let the two parallel flraight lines g h, l m touch 
the ellipfe a d b e, or oppofite hyperbolas f b n, iak 
in the points b, a, and let the ftraight line a b, joining 
the points of contact, be bife&ed in c ; the point c is 
the center of the ellipfe or oppofite hyperbolas, and no 
other point, befides c, can be the center of the ellipfe 
or oppofite hyperbolas. 

Part I. Take any other point n in the curve of the 
ellipfe, or in the curve of either of the oppofite hyper¬ 
bolas. Then n c being drawn, and produced, it will 
meet the curve of the ellipfe, or the curve of the oppo¬ 
fite hyperbola, and the whole line terminated by the 
curve, or curves, will be bife&ed in c. For let n c be 
not parallel to g h, l m, and draw n o f parallel to 
them, and let it meet a b in o, and the curve again in 
F. Make a s equal to b o, and through s draw isk 
parallel to the tangents g h, l m, or to n o f, and let 
-K be one of the points in which it meets the curve. 
Then, by Prop. I. n f will be bife&ed in o, and s K 
will be half the whole line, of which it is a part, ter¬ 
minated by the curve; and therefore, by Prop. XIII. 
Book I. A o X o B : n o 2 : : b s x s a : k s 2 . But as 
A s, b o are equal to one another, a o X ob is equal to 
bs X s A, and therefore (14. v.) the fquare of n o is 
E equal 


49 


BOOK 

II. 


Fig. 37* 
38 . 



56 OF THE ELLIPSE AND HYPERBOLA* 

BOOK equal to the fquare of k s, and no is equal to K S. 

llf Let n c produced meet K s I in I, and then as c s, c o 
* are equal, and (29. i.) the triangles nco, i c s equi¬ 

angular, it follows (26. i.) that n c is equal to c 1, and 
1 s is equal toNo; and therefore, by the above, 1 s is 
equal to s k. Confequently, by Prop. I. the point 1 is 
in the curve, and therefore n c being produced, it 
meets the curve of the ellipfe again, or the curve of 
the oppofite hyperbola, and the whole line n 1, termi¬ 
nated by the curve, or curves, is bife&ed in e. The 
point c is therefore the center of the ellipfe or oppofite 
hyperbolas; for in the ellipfe the ftraight line parallel 
to the tangents G h, l m, and paffing through c is alfo 
bife&ed in c, by Prop. I. 

Part II. No other point, befides c, can be the center 
of the ellipfe, or oppofite hyperbolas. In the ellipfe 
this is evident; for if there could be another, then a 
ftraight line paffing through c and that other center, 
and terminated by the curve, would, by the fecond 
Definition, be bife&ed in two points : which is abfurd. 
Fig. 38. Nor can the hyperbolas fbn,iak have any other 
center befides c. For through c draw d e parallel to 
G h, l m ; and fuppofe the point d in this ftraight line 
to be another center. Through d draw the ftraight 
line f 1 parallel to a b, and let it meet the tangents in 
G, l, and the oppofite curves in f, 1. Then, by Cor. 
Prop. I. f g is equal to 1 l; and (34. i.) as g d is equal 
to b c, and d l to c a, and b c equal to c A, it follows 
that f d, d 1 are equal. But through d draw a ftraight 
line o, t, not parallel to a b, and let it meet the curves 
s in o, t 5 which it evidently may do, by Prop. IX. 
Book I. as it may be drawn parallel to a ftraight line 
drawn from a point in the curve fbn through c and 
meeting the oppofite curve, by Part I. Let q t meet 
the ftraight lines fn,ik parallel to the tangents g h-, 

l M, 



t)P THE EtLFP&E AND HYPERBOLA, 

i M, in r and p. Let r be between the points f, n, 
and then p will be without i, k, or without the hyper¬ 
bola i a k. Then, by the above, as f d is equal to 
D i, and (29. i.) the triangles fdr,idp equiangular, 
it follows (26. i.) that it d is equal to pd. Confe- 
quently t q is not bife&ed in d, and therefore d is not 
a center. Nor can any point out of the line d e be a 
center; for if it could, then a ftraight line drawn 
through it, and parallel to A b, and meeting the curves, 
would, by the above, be bife&ed by d e, and, according 
to the firft Definition, it would alfo be bife&ed in this 
other center. The fame ftraight line would therefore 
be bifefted in two points: which is abfurd. Confe- 
quently no other point befides c can be a center. 

Cor. 1. A ftraight line, as a b, joining the points of 
.contact of two parallel tangents G h, l m, is a dia¬ 
meter of the ellipfe, or oppofite hyperbolas ; and d e 
drawn through the center c, and parallel to the tan¬ 
gents, or to the fecants f n, i k parallel to them, is 
alfo a diameter. For in the ellipfe d e is a diameter, 
according to the fecond Definition ; and in the hyper¬ 
bola d e is a fecond diameter, by the third Definition, 
as it bife&s, by the above, any ftraight line parallel to 
a b, and terminated by the curves of the oppofite hy¬ 
perbolas. 

Cor. 2. If two ftraight lines a l, b g touch an el- 
lipfe, or oppofite hyperbolas in a, b, the vertices of the 
diameter a b, they will be parallel. For if b g be not 
parallel to A l, the tangent parallel to a l will meet 
the curve of the ellipfe, or the curve of the oppofite hy¬ 
perbola, not in b, but in fome other point, as n, as two 
ftraight lines cannot touch a conic fedtion in the fame 
point. Then, by the preceding Cor. if a n be drawn, 
it will be a diameter: which is abfurd,. For the 


5 * 


BOOK 

II. 


Fig. 37. 
38. 



5 * 


OP THU ellipse and hyperbola. 


BOOK 

u: 


Fig. 37. 
38 . 


Fig. 38. 


Fig. 39- 
40. 


ftraight line a n cannot pafs through c, the center, as 
c is in the diameter ab, 

prop. nr. 

If two parallel ftraight lines touch an ellipfe or oppofite hy~ 
perbolas , ftraight lines parallel to them in the ellipfe , or 
in either hyperbola , will be ordinates to the diameter 
joining the points of contact; but , in the oppoftte hyper - 
bol as, ftraight lines parallel to the diameter joining the 
points of contaSl will be ordinates to the fecond diameter 
parallel to the tangents', and, in either cafe , ordinates 
to the fame diameter of an ellipfe , or oppofte hyperbolas , 
are parallel to one another . 

Parti. Let two parallel ftraight lines g h, l m touch 
the ellipfe a d b e, or oppofite hyperbolas fbn, iak 
in the points a, b, and let f n, i k, in the ellipfe or in 
either hyperbola, be parallel to gh, l m ; then a e 
will be a diameter by Cor. 1. Prop. II. and by Prop. I. 
it will bifeft fn, i k. This part is therefore evident, 
by the above and the fourth Definition. 

Part II. The reft remaining as above, let d e be a 
fecond diameter, parallel to the tangents g h, l m, 
and let fi,nk } terminated by the oppofite curves, be 
parallel to a b, and then by Cor. 1. Prop. II. fi,nk 
are bife&ed by d e, and are therefore ordinates to it, 
according to the fourth Definition. 

Part III. Ordinates to the fame diameter of an el¬ 
lipfe, or oppofite hyperbolas, are parallel to one an¬ 
other. For firft let a b be any diameter of the ellipfe, 
or any tranfverfe diameter of the‘Oppofite hyperbolas ; 
and c being the center, let l a, g e be the parallel 
tangents drawn through the vertices a, b, according to 
Cor. 3. Prop. II. Then, by Part I. any ftraight line 

parallel 



OP THE ELLIPSE AND HYPERBOLA* ^3 

parallel to l a, or g b in the'ellipfe, or in either of the BOOK 
oppofite hyperbolas, will be an ordinate to a b. But, 
if it be poffible, let the ftraight line i p in the ellipfe, 
or in either of the oppofite hyperbolas, be an ordinate 
to the diameter a b, and not be parallel to l a, g b. 

Draw i k parallel to l a or g b, and let it meet a b in 
s, and the curve in k. Draw i c, and, being pro¬ 
duced, let it meet the curve of the ellipfe again, or the 
curve of the oppofite hyperbola, in n ; and draw k n. 

Then, by Part I. i K is bifefted in s; and, as i n is bi- 
fe&ed in c, i c : c n : : i s : s k, and therefore (2. vi.) 
ab, k n are parallel. Confequently, if 1 p meet a Bin r 
and k n in v, (2. vi.) is:sk::ir;rv, and there¬ 
fore i v is bife6ted in r. But the ftraight line k n in 
Fig. 39. is wholly within the ellipfe, and in Fig 40. 
n k is without the oppofite hyperbolas, and being pro^ 
duced it falls on the one fide within one hyperbola, 
and on the other within the oppofite hyperbola. The 
other point p therefore, in which 1 p meets the curve, 
cannot be in k n, and confequently 1 p cannot be bir 
fe£ted in r, or be an ordinate to ab. 

Laftly, the reft remaining as above, let d e be a fe- Fig. 40, 
cond diameter of the hyperbolas parallel to the tan^ 
gents la, g b. Then any ftraight line parallel to a b, 
and meeting the oppofite curves will be bife&ed by 
D e, according to Part II. But, if it be poflible, let 
1 x meet the oppofite curves in 1, x, the diameter d e 
in y, and be an ordinate to d e, and not be parallel to 
a b. Let 1 f be parallel to ab and meet the oppofite 
curves in 1, f, and de in d. Draw 1 c, and, being 
produced, let it meet the oppofite curve in n. Draw 
N f, and let it meet 1 x in z. Then, as 1 n is bife&^d 
in c, and as 1 f, according to Part II. is bife6ted in d, 
jc:cn::id:df, and therefore (2. vi.) N F is pa- 
E 3 rallel 



54 OP THE ELLIPSE AND HYPERBOLA. 

BOOK rallel to d e. Conlequently (2. vi.) id:df::iy: 

IL y z, and therefore i y is equal to y z. The ftraight 
~~ line i x therefbre cannot be bife&ed in y, or be an or¬ 
dinate to d e. In every cafe therefore, ordinates to 
the fame diameter are parallel to one another. 

Cor. i. If a ftraight line bife& two parallel ftraight 
lines in an ellipfe or hyperbola, or oppofite hyperbolas, 
it will be a diameter : and ftraight lines drawn through 
its vertices, and parallel to the lines bife&ed, will touch 
the ellipfe, or oppofite hyperbolas, if in the oppofite hy¬ 
perbolas it be a tranfverfe diameter. For by Cor. 2. Prop. 
VIII. Book I. two ftraight lines, and only two, parallel 
to the lines bife&ed, can be drawn to touch the ellipfe 
or oppofite hyperbolas; and by Cor. 1. Prop. II. the 
ftraight line joining the points of conta& is a diameter, 
and, by this Propofition, this diameter will bifeft only 
fuch ftraight lines in the ellipfe or oppofite hyperbolas 
as are parallel to the tangents. It is alfo evident from 
this Propofition that a ftraight line bifecting two parallel 
ftraight lines, terminated by the curves of oppofite hy¬ 
perbolas, is a fecond diameter of the hyperbolas. 

Cor. 2. If in an ellipfe, hyperbola, or oppofite hy¬ 
perbolas, a diameter bife& a ftraight line not palling 
through the center, it will alfo bife& any line parallel 
to it in the fame fe&ion or oppofite hyperbolas. For, 
by the preceding Cor. a ftraight line bile&ing two pa¬ 
rallel ftraight lines in an ellipfe, hyperbola, or oppofite 
hyperbolas, is a diameter, and therefore paftes through 
the center. Confequently a diameter, or a ftraight line 
palling through the center, and bife&ing one of two 
parallel lines in the fame fe&ion, or oppofite hyper¬ 
bolas, will alfo bife£t the other. 

Cor. 3. A diameter of an ellipfe or hyperbola will bi- 
fe£t all ftraight lines in the fedlion parallel to a tangent 

palling 



OF THE ELLIPSE AND HYPERBOLA. 

paffing through its vertex; and ordinates to a diameter 
and tangents paffing through its vertices are parallel to 
one another. 


PROP. IV, 

Two diameters of an ellipfe , or oppojite hyperbolas, are 
conjugate diameters , if one of them he parallel to the or- 
dinates of the other . 

Let AEjDE, be two diameters of the ellipfe A d b e, 
or of the oppolite hyperbolas m a, g b f, and let the 
diameter de be parallel to gf a double ordinate to 
a b ; the diameter d e will be the conjugate diameter 

to A B. 

For let c be the center, and f c being drawn, and 
produced, let it meet the curve of the ellipfe, or the 
curve of the oppofite hyperbola in m. Draw g m, and 
let it meet d e in p. Let a b meet g f in h, and then 
as g f is a double ordinate to the diameter a b, it is bi- 
fedted in h ; and as the diameter F m is bife&ed in c, 
the center, fc:cm ::fh: hg, and therefore (2. vi.) 
m g is parallel to a b. Again, asDE, gf are parallel, 
(2. vi.) m c : c f : : m p : p g, and therefore m g is 
bifedted in p. Confequently the diameters de, ab 
are conjugate to one another, according to the fixth 
Definition; for by Cor. 2. Prop. III. d e will bifedt 
any ftraight line parallel to m g or a b, and A b will bi- 
l'edl any ftraight line parallel to G F or d e, the ftraight 
lines parallel to m g or g f being terminated by the 
curve of the ellipfe, hyperbola, or oppofite hyperbolas. 

Cor . 1. From the above. Cor. 3. Prop. III. and 
Cor. j . Prop. II. it is evident, that if two parallel 
ftraight lines touch an ellipfe or oppofite hyperbolas, 
and if, from any point in the curve of the ellipfe, or of 
either hyperbola, except the points of contact, a ftraight 
e 4 line 


55 


BOOK 

II. 


Fig. 41 „ 
4 2 * 



56 


BOOK 

II. 


Fig. 4:. 


OF THE ELLIPSE AND HYPERBOLA. 

line be drawn parallel to the tangents, it will be either 
an ordinate to the diameter joining the points of con¬ 
tact, or in the ellipfe it will be the diameter conjugate 
to that joining the points of conta&. 

Cor. 2. Ordinates to a diameter, of an ellipfe or op- 
pofite hyperbolas, tangents pafling through its vertices, 
and its conjugate diameter are parallel to one another. 

DEFINITIONS. 

VIII. 

If c be the center of the oppofite hyperbolas m a, f b, 
and a b a tranfverfe diameter, to which d e is the con¬ 
jugate, and h f an ordinate, and if the rectangle under 
A h, h b be to the Iquare of 11 f as the fquare of cb to 
the fquare of c e or c d, the points d, e are called the 
Vertices of the fecond 'Diameter de. In this way the 
magnitude of any fecond diameter is determined by its 
vertices. 

Cor. If the ordinate h f be parallel to the bafe of 
the cone, in which the hyperbola was formed, the fe- 
midiameter c e or c d will be equal to the ftraight line 
drawn through c and parallel to the bafe of the cone, 
and touching either of the conical fuperficies. For, in 
this cafe, by the fecond Cor. to Prop. X. Book I. the 
re&angle under a h, h b is to the fquare of h f as the 
Iquare of cb to the fquare of the line drawn through 
c and parallel to the bafe of the cone, and touching ei¬ 
ther of the conical fuperficies ; and, by this Definition, 
the fquare of c b is to the fquare of c e or d e in the 
fame proportion. Confequently (9. v.) the Cor. is evi¬ 
dent, as the tides of equal fquares are equal. 

. IX. 

A ftraight line which is a third proportional to two 
conjugate diameters of an ellipfe, or oppofite hyper¬ 
bolas. 



1 'Ul/f \ f//KH/r .'>(> 



J ftasire sc. 

























































OF THE ELLIPSE AND HYPERBOLA* 57 

bolas, is called the Parameter, or Latus Pedum, of that BOOK 
diameter which is the firft of the three proportionals. 1L 

PROP. V. 

If each of two ftraight lines, meeting one another, touch or 
cut, or one of them touch, and the other cut, an ellipfe, 
hyperbola, or oppofite hyperbolas ; the fquare of the frjl 
of the two, if a tangent, or the red;angle under its feg- 
•nunts, if a fecant, will be to the fquare of the fecond, if 
a tangent, or the redangle under its fegments, if a fe¬ 
cant, as the fquare of the femidiameter parallel to the 
frjl to the fquare of the femidiameter parallel to the fe¬ 
cond\ 

In the ellipfe, and when the two_ ftraight lines are 
parallel to two tranfverfe diameters of oppofite hyper¬ 
bolas, the Propofition is evident from Prop. XIII. 

Book I. For diameters in the ellipfe, and tranfverfe 
diameters of oppofite hyperbolas, are fecants meeting 
one another in the center, in which they are bifefted. 

For other cafes, firft let l m , g h cutting either of Fig. 43* 
the oppofite hyperbolas in l, m and g , h, meet one 
another in k ; and let c e be the femidiameter parallel 
to l m , and let c f be the femidiameter parallel toGH, 
and let c e, l m be parallel to the bafe of the cone in 
which the hyperbola was formed ; and then l k X 
km : g k X k h : : c e 2 : c f 2 . For, bifeft G h in 1, 
and through 1 draw n p parallel to lm or ce, and 
draw the tranfverfe diameter iba, Then, by Cor. 2. 
to Prop. X. Book I. and the Cor. to Definition VIII. 
n 1 x 1 p : a 1 X 1 b : : c e 2 : c b 2 ; and by Definition 
VIII. A 1 x i b : i h 2 : : cb 2 : c f 2 . Confequently, 
n 1 X 1 p : A r X 1 B : 1 H 1 
ce 2 : c b 2 : c f 2 , . 

and (22. v.) N 1 X I p : 1 H 2 : : c E 2 : c F 2 . But, by 

Prop. 



58 


BOOK 

II. 


Fig. 44 . 


OF THE ELLIPSE AND HYPERBOLA, 

Prop. XIII. Book I. nixip:ih*::lkxkm: 
g k x k h ; and therefore (11. v.) l k X k m : g k X 
K H : : c e z : c f\ 

Secondly, let g h, l m cut one another in k, in the 
hyperbola as above, and let neither of them be parallel 
to the bafe of the cone in which the fe&ion was 
formed. Let g h be parallel to the femidiameter c f, 
and let l m be parallel to the femidiameter c e. Through 
K draw a r parallel to the bafe of the cone in which 
the fe&ion was formed, and let c s be the femidiameter 
parallel to a r, or to the bafe of the cone. Then, as 
above, we have the two following ranks of magnitudes 
proportionals, 

LK X K M : QK XKRIGKXKH 
ce : : c s 3 : c f 2 ; 

‘and therefore (22. v.) l k x k m : g k x k h : : c e 2 : 
t: f 2 . 

Thirdly, the reft remaining as above, let the ftraight 
line v t k meet one of the oppofite hyperbolas in v, 
the other in t, and the ftraight line a r in k. Then, 
by Prop. XII. Book I. and Cor. to Definition VIII. 
y x being the tranfverfe diameter parallel to v t, y k 
x k t : q k X k r : : c x 3 : c s 2 . 

Laftly, every thing remaining as in the two preced¬ 
ing- cafes, by the above we have the two following 
ranks of magnitudes proportionals, 

vkx kt:okxkr:gk xkh 

c x 2 : c s 2 : cf 2 , and there¬ 

fore (22. V.) V K X K T : G K X K h : : c x 2 : C F 2 . 

In every cafe therefore, by the above, and Prop. XIII. 
Book I. (and j 1. v.) u If each of two ftraight lines,” 
&c. 

Cor . 1. If two ftraight lines be ordinates to any dia¬ 
meter of an ellipfe, or tranfverfe diameter of an hyper¬ 
bola, the fquare of the firft will be to the fquare of the 

fecond. 



OF THE ELLIPSE AND HYPERBOLA. 

fecond, as the re&angle under the abfciffes correfpond- 
ing to the firft to the rectangle under the abfciffes cor- 
refponding to the fecond. 

Cor. %. From this Propofition (and 22. vi.) it is evi¬ 
dent, that if two ftraight lines meeting one another 
touch an ellipfe, hyperbola, or oppofite hyperbolas, 
they will be to one another as the femidiameters to 
which they are parallel. 

Cor . 3. From this Propofition, and the firft Lemma, 
it is evident, that if two conjugate diameters of an el¬ 
lipfe cut one another at right angles, they cannot be 
equal to one another; for if they were equal to one 
another, the fection would be a circle, by the firft 
Lemma, as the fquare of the ordinate would be equal 
to the rectangle under the correfponding abfciffes. 

PROP. VI. 

If a ftraight line be an ordinate to any diameter of an el - 
lipfe , or any tranfverfe diameter of an hyperbola , the 
redangle under the abfciffes of the diameter will be to 
the fquare of the ordinate as the diameter to its para¬ 
meter . 

Let the ftraight line g f be an ordinate to the dia¬ 
meter a b of the ellipfe b g, or to the tranfverfe dia¬ 
meter a b of the hyperbola b g, and let b h be the pa¬ 
rameter of a b ; the re&angle under the abfciffes A f, 
f b is to the fquare of f g as A b to b h. 

Let c be the center of the ellipfe or hyperbola, and 
let d e be the diameter parallel to g f, and confe- 
quently, by Prop. IV. the conjugate diameter to A b. 
Then, by the ninth Definition, abide:: d e : b h ; 
and therefore (Cor. 2. 20. vi.) ab 2 :de 2 :: abibh, 
But (15. v.) a b 2 : d e 2 : : c b 2 : c D 2 ; and therefore 
^ri. v.j c b 2 : c D z : : a p : b H. But, by Prop. V. 

c B* 


59 

BOOK 

II. 


Fig- 4<* 
46 . 



6 o 


OF THE ELLIPSE AND HYPERBOLA. 


BOOK 

II. 


CB 1 : c d 2 : : A f X fb : fg 1 ; and confequently 
(u. v.) afxfb:fg 2 ::ab:bh. 

Cor. i. Let the parameter b h be at right angles to 
the diameter a b, and from the other vertex A, draw 
a h. From the point f draw f k perpendicular to a b, 
and let it meet A H, or A h produced, in k. Complete 
the rectangle k b, and it will be equal to the fquare of 
the ordinate f g. For, as e h, f k are at right angles 
to A B, they are parallel, and therefore (4. vi.) A b : b h 
::Af:fk; and confequently, by this Propofition 
(and 11. v.) A f X f b : f g 2 : : A f ; f k. But (1. vi.) 
A f : f k : : A f x fb:fkxfb, and therefore a f 
xfb:fg 2 ::afxfb: fkxfb. Confequently 
(14. v.) f k x f b is equal to f g 2 . 

Cor. 2. Complete the rectangle l a b h, and let l h 
meet f k in m, and let k n, the fide of the re£bangle 
k b, oppofite to b f, meet b h in n ; then in the ellipfe 
the fquare of the ordinate f g is lefs than the rectangle 
under the abfcifs f b and the parameter b h, by the 
re&angle m n, fimilar to l b, and having one of its 
fides equal to b f ; but in the hyperbola, the fquare of 
the ordinate f g, is greater than the re<$langle under 
the abfcifs b f and the parameter b h, by the rectangle 
m n fimilar to l b, and having one of its fides equal to 
b f. This is evident from the preceding Cor. 

SCHOLIUM. 

On account of the deficiency of the fquare of f g 
from the re6fangle under f b, b h in Fig. 45. Apollo¬ 
nius called the fe&ion an ellipfe ; and on account of 
the excefs of the fquare of f g above the rectangle un¬ 
der f b, b H in Fig. 46. he called the feftion an hy¬ 
perbola. 

From the properties demonftrated above thefe fec- 

tions 




OF THE ELLIPSE AND HYPERBOLA. 


tions are frequently denoted by Algebraical equations, 
in the following manner. Put the diameter a b (iu 
Fig. 45. and 46.) = a , its parameter b h == p } the ab- 
lcifs f b = x, and the ordinate f g =jy. Then a f = 
a~^ x, the negative fign applying to the ellipfe, and 
the pofitive fign to the hyperbola. And by the fimilar 

. , — ap ~ px 

triangles ABH, A f k, a \ p : : a + x : - ■ ■ ■■ = fk. 

Confequently by the firftCor. to Prop. VI. — .X x 




PROP. VII. 

If a Jlraight line, touching an ellipfe or hyperbola , meet a, 
diameter , and from the point of contaci there be drawn 
an ordinate to the diameter; the fe?nidiameter will be 
a mean proportional between the fegments of the diame¬ 
ter } between the center and ordinate , and between the 
center and tangent . 

Firfl-, let the ftraight line e m, touching the ellipfe 
or hyperbola e i g in the point e, meet any diameter 
a 1 in the ellipfe or tranfverfe diameter of the hyper¬ 
bola in m, and let e f be an ordinate to the diameter, 
and meet it in f, and let c be the center; the femi- 
diameter c 1 is a mean proportional between the feg¬ 
ments c f, c M. 

For, let a b, 1 d be tangents pafling through the 
vertices a, i, and meeting the tangent e m in b and d. 
Then, by Cor. to Prop. XIII. Book I. e b : e d : : 
a E : 1 d ; and as, by Cor. 3. Prop. III. the ftraight 
lines a b, e f , id are parallel, it is evident (from io. 
vi.) that e B : e D : : a f : 1 f . And as (29. i.) the 
triangles b a w, dim are equiangular, a b : 1 d : : 

a m 


61 


BOOK 

ji. 


Fig. 47- 
48 . 





6 % 


OK THE ELLIPSE AND HYPERBOLA^ 


BOOK 

II. 


Pig. 49. 


Fig. 47 . 

48. 


A M : I m. Hence (11. v.) A m f I m : : a f i I F* 
and (i8.v.)am + im:im::af + if:if; and 
by halving the antecedents it will be in the ellipfe 
c m : 1 M : : c 1 : 1 Fj but in the hyperbola c 1 : 1 m ; : 
c f : if. Confequently, by converlion, it will be in 
the ellipfe cm:ci:;ci:cf; but in the hyperbola 
C 1 : c m : : c f : c 1, and therefore cm:ci::ci: 

C F. 

Let a 1 be now a fecond diameter of the oppoli to 
hyperbolas gk,el and e f an ordinate to it; and let the 
tangent e m meet it in m, and the tranfverfe diameter 
K l, parallel to e f, in n. Then, by Prop. IV. kl,ai 
are conjugate diameters; and therefore ep being drawn 
parallel to A 1, and meeting k l in p, it will be an or¬ 
dinate to k l. Let c be the center, and then, by the 
above, cp:cl::cl:cn, and therefore (Cor. 2. 
20. vi.) c p 2 : c l 2 :: c p : c n. But (34. i.) cp, fe 
are equal, and F. p is equal to f c, and (4. vi.) e f j 
c n : : M f : m c, and therefore c p 2 : c L 2 : : m f : 

m c. Hence (17. v. and 6. ii.) k p x p l ; c l 2 : ; 

c F : m c; and (1. vi. and 11. v.) k p X p l ; c l 2 : : 

c f 2 : c F X m c; and (16. v.) k p x p l : c f 2 or 

E p 2 : : c l 2 : c F x m c. But, by Def. viii. k p X 
p L : e p 2 : : c l 2 : c i 2 , and therefore (11. and 9. v.) 
c F X m c is equal to c 1% and confequently c m : 
c 1 : : c 1 : c f . 

Cor. 1. From the above (and 37. vi.) cm X c f is 
equal to c i 2 , and in the ellipfe thefe equals being 
taken from c m 2 , we have (6. ii v and 2. ii.) a m x 

m 1 equal to c m x m f. But, when a 1 is a tranf¬ 

verfe diameter in the hyperbola, c m 2 being taken from- 
the equals, cm x c f, c i 2 , we have (5. ii. and 3. ii.) 
a m x m 1 equal to c m x m f. 

Cor. 2. As c m x c f is equal to c I 2 , by taking 

from each c f 2 in the ellipfe, we have (3. and 5. ii.) 

c F 



OP THE ELLIPSE AND HYPERBOLA. 6 $ 

t f X f m equal toAPXFi. But a i being a tranf- BOOK, 
verfe diameter in the h}q:>erbola, by taking the equals IL 
c m x c f, c i z from c f 2 , we have ( 3 . and 6 . ii.) 
c f x f m equal to a f x f i. 

Cor. 3. When A 1 is a diameter of the ellipfe or tranf- 
verfe diameter of the hyperbola, by the demonflration 
of the fir ft part of the Propofition, a m : 1 m : : a f ; 

FI. 


PROP. VIII. 

If two Jlraight lines , touching an ellipfe , hyperbola , or op- 
pojite hyperbolas , meet one another , the diameter bifed- 
ing the line joining the points of contad will pafs through 
the point of concourfe . 

Let the two ftraight lines e m, g m touch the el- Fig. 47* 
lipfe or hyperbola e i g, or the oppofite hyperbolas e l, 
g k, in the points e, g, and meet one another in m, 
and let the diameter c f bile£t e g, the ftraight line 
joining the points of contadl in f ; the diameter c F 
will pafs through m. 

For let c be the center, and let A, 1 be the vertices 
of the diameter; and then, as e g is bife&ed by the 
diameter c f, e f is an ordinate to it, and therefore, by 
Prop. VII. c f : c 1 : : c 1 : the fegment of the dia* 
meter intercepted between c and the tangent e m . For 
the fame reafons c f : c 1 : : c 1 : the fegment of the 
diameter intercepted between c and the tangent g m. 
Confequently the fegment of the diameter between (t 
and the tangent e m, is equal to the fegment of the 
diameter between c and the tangent g m. The dia¬ 
meter muft therefore pafs through m ; for if it did not, 
it would, upon being produced, firft meet the one tan¬ 
gent, and then the other, and its fegments between c 
and the tangents vvpuld be unequal. 


Cor, 




6 4 


OP THE ELLIPSE AND HYPERBOLA. 


BOOK Cor . If two flraight lines touching an ellipfe, hyper- 
1L bola, or oppofite hyperbolas, meet one another, a flraight 
" line pafling through the point of concourfe, and bifeft- 

ing the line joining the points of contaft, will be a 
diameter. 


PROP. IX. 

If tzuo parallel flraight lines touching an ellipfe , or op- 
pofte hyperbolas, meet a third tangent , the reft angle un - * 
der their fegments , between the points of contact and the 
points of concourfe , will be equal to the fquare of the fe- 
midiameter to which they are parallel ; and the redan¬ 
gle under the figments of the third tangent , between its 
point of contad and the parallel tangents , will be equal ; 
to the fquare of the femidiameter to which it is parallel . 1 

Pig. 5o . Let the parallel flraight lines ab, id touch an el- 
lipfe e i, or oppofite hyperbolas A, i, in the points a, i, 
and let them meet in b, d a flraight line B d, which 
touches the ellipfe, or one of the oppofite hyperbolas 
in e ; then the rectangle under the fegments a' e, id 
is equal to the fquare of the femidiameter parallel to 
A b or i d, and the rectangle under the fegments b e, 
e d is equal to the fquare of the femidiameter parallel 

to B D. 

For let c be the center, and draw e g parallel to 
ab, id; and draw alfo a i. Then by Cor. i. to 
Prop. II. a i is a diameter, and, by Cor. i. to Prop. 
IV. e g is either an ordinate to a i, or in the ellipfe 
the conjugate diameter to it. 

rig 50 ' Firfl, let eg be the conjugate diameter to a i, and 
then by Prop. IV. and Cor. 3. Prop. III. bd,ai are 
parallel to one another, and alfo a e, c e, i d to one ■ 
another. Confequently (34. L)ab,ce,id are equal 
to one another, as are alfo ac,ci,be, e d to one 

another; j 



PltUe \in.page 0 / 



J.Bastre sc. 















































































OP THE ELLIPSE AND HYPERBOLA-' 





be x e d is equal to A c 2 . 

Next, let e g be an ordinate to the diameter A I, and pi gt 
let it meet it in f . Let c k be the femidiameter pa- 5** 

rail el to the tangents a b, i d, and c p the femidia¬ 
meter parallel to the tangent s b ; and let e b meet the 
diameter a i in m, and the diameter k c l in h. Let 
E o be an ordinate to k c l, and let it meet it in o. 

Then, by Cor. 3. Prop. III. and Prop. IV. a b, l h, 
g e, 1 d are parallel, and e o is parallel to A 1. By 
Cor. 1. Prop. VII. A m x m i is equal to c m x m f, 
and therefore (16. vi.) A m : c m : : m f : m i; and 
(4. vi.) A m : c M : : A b : c H, and m f : m i : : f E or 
c o : 1 d. Confequently (ii.v.) ab:ch : : c o : 1 d, 
and (16. vi.) A b X I D is equal to c h x c o, which 
is equal to c k 2 , by Prop. VII. (and 17. vi.) and there¬ 
fore a b x 1 d is equal to c k 2 . 

Again, by Cor. 2. Prop. V.ab:be::ck:cp, 
and by Cor. to Prop. XIII. Book I. ab : id : : be : 
e d. Hence (22. vi.) abxid:bexed::ck 2 : 
c p 2 , and as by the above a b x 1 d is equal to c k 2 , 
it follows (14. v.) that b e X e d is equal to c p 2 . 

Cor. 1. Every thing remaining as above, as, by 
Cor. 2. Prop. VII. a f X f 1 is equal to c f X f m, 

(16. vi.) A F : f m : : c f : f 1. But on account of the 
parallels (10. vi.) af:fmj:be:em; and c f : 
f 1 : : h E : E D. Confequently (11. v.) b e : e m : : 
it e : e d, and therefore (16. vi.) m e X e h is equal 
to b e X e d or to c p 2 . 

Cor . 2. If a liraight line as b e, touching an ellipfe 
or hyperbola in the point e, meet the diameter a 1 in 
m, and the diameter k l in h, and if the rectangle un¬ 
der m e, E H be equal to the fquare of the femidia¬ 
meter parallel to b e, the diameters a I, K l will be 
conjugate to one another. 


F 


PROP. 



66 


OF THE ELLIPSE AND HYPERBOLA* 


BOOK 

II. 


PROP. X. 


To find the axes of a given ellipfe or hyperbola , the center 
being alfo given ; and to demonjlrate that the fame fec- 
lion can have only two axes . 


Fig. 53. Part I. Let d b e be a given ellipfe or hyperbola, of 
54 * which c the center is alfo given \ it is required to find 
the axes of the fe&ion. 

Fig. 53. In the ellipfe draw cg,cf two femidiameters, and, 
if they be unequal, let c c be greater than c f. With 
c as a center, and a diftance lels than c g but greater 
than c v, deferibe the circle h d e. Then from this 
conftru&ion and the nature of the two curves it is evi¬ 
dent, that the circumference of the circle will cut the 
curve of the ellipfe in four points, two of them being 
towards the left of the center, as the figure is viewed, 
and two of them towards the right. Let the circum¬ 
ference of the cirele cut the curve of the ellipfe in the 
points d, e. Draw the ftraight line d e; and through 
C draw a b bife&ing D e in 1, and (3. iii.) a b will be 
at right angles to d e. Through c draw l m parallel 
to d e, and a b, l m will be the axes of the ellipfe. 

For, by conftru£tion, de is an ordinate to the dia¬ 
meter a b, and at right angles to it. Again, as l m is 
parallel to d e, and as a i d is a right angle, the angle 
acl (29. i.) is alfo a right one. By Prop. IV. the 
diameter l m will alfo bile£ all ftraight lines in the 
ellipfe parallel to a b, and it will therefore cut its ordi¬ 
nates at right angles. Confequently a b, l m are axes 
of the ellipfe, according to the feventh Definition. 

If the femidiameters c g, c f be equal, then a dia¬ 
meter bife&ing the angle gcf will be one of the axes, 
and a diameter at right angles to it will be the other. 

I or, in this cafe, if g f be drawn, it will be bifedled at 

right 



OE THE ELLIPSE AND HYPERBOLA. 6j 

right angles (4. i.) by the diameter bife&ing the angle book 
gcf; and the reft will be as above. Ir - 

Next, let the fe&ion d b e be an hyperbola, of which Fjcr c ~ 
c is the center, and let k be any point within the hy¬ 
perbola. With c as a center, and c k as a diftance, 
defcribe the circle e k d, and let its circumference cut 
the curve of the hyperbola in the points e, d. Draw 
d e and bife& it in I, and through 1 draw the diame¬ 
ter a e ; and parallel to de draw the diameter l m. 

The diameters A b, l m are the axes of the hyperbola. 

For d e is a double ordinate to the diameter a b, and 
(3. iii.) a b cuts it at right angles, and l m is parallel 
to the ordinate d e. 

Part II. To deinonftrate that an ellipfe or hyperbola 
can have only two axes. Firft, let the fe&ion b d a be Fig. 55 , 
an ellipfe, and c being the center, let r l, f g be the 
axes, found as above; and, if it be poffible, let the dia¬ 
meter a b be alfo an axis. Let l d be a double ordi¬ 
nate to A b, meeting it in e, and the curve again in d ; 
and the diameter d k being drawn, let d h be an or¬ 
dinate to r l *. Then, as by hypothecs A b is an axis, 
c e l, c e d are right angles, and as l d is bife&ed in 
e, (4. i.) c l is equal to c d. Again, as by the above 
R l, f g are conjugate, dh is parallel to f g, by Prop. 

IV. and by Prop. V.cl 2 :cf 2 :: rh X h l : d h 2 . 

But, by Cor. 3. Prop. V. c l, c f muft be unequal, and 
therefore, fuppofing cl to be the greateft, c l 2 is 
greater than c f 2 , and r h X h l greater than d h 2 . 

To thefe unequals add c H 2 , and then (5. ii. and 47. i.) 
c l 2 is greater than c d 2 ; and confequently c l is 
greater than c D. But c d is alfo equal to c l : which 


* A method of drawing an ordinate to a given diameter will be In- 
ferted hereafter. The infertion of it previous to the above, or at this 
place, would have caufed a needlefs repetition. 

F % is 



68 


OF THE ELLIPSE AND HYPERBOLA. 


BOOK 

II. 


Fig* 56. 


Fig. 55- 


is abfurd. The diameter a b therefore cannot be an 
axis. 

Next, let the fe&ion e b d be an hyperbola, of 
which c is the center, f a the oppofite hyperbola, and 
a b, l M the axes found as above ; and, if it be poffi- 
ble, let the tranfverfe diameter df be an axis. Let 
d t touch the hyperbola in d, and meet the axis a b 
in t ; and let die be an ordinate to A B, and let it 
meet it in 1. Then in the triangle c 1 d, c i d is a 
right angle, by Def. VII. and therefore cdi is lefs 
than a right angle, and confequently cdt is much 
lefs than a right angle. But, by Cor. 2. Prop. IV. the 
tangent t d is parallel to the ordinates of the axis f d, 
and therefore, by Def. VII. (and 29. i.) the angle tdc 
is a right one. And, by the above, it is alfo lefs than 
a right one: which is abfurd. Confequently df is 
not an axis. Nor can a fecond diameter as g h, befides 
l m, be an axis. For fd conjugate to g h being 
drawn,and dt a tangent, the demonftration would end 
in the fame abfurdity. 

Cor. From the above, and Prop. IV. it is evident, 
that the axes of an ellipfe, or hyperbola, are conjugate 
diameters. 


PROP. XI. 

Of all the diameters of an ellipfe the greater axis is the 
greatejl and the lejjer axis is the leaf ; and of oppofte 
hyperbolas the axes are the leaf diameters. 

Part I. Let a b d be an ellipfe, of which c is the 
qenter, r l the greater axis, and f g the lefler axis ; of 
all the diameters r l is the greateft and f g the leaft. 

For let k d be any other diameter, and let d h be 
an ordinate to r l, and d m an ordinate to f g. Then, 
by Cor. 2. Prop. IV. d h is parallel to f g, and d m to 


r l: 



OF THE ELLIPSE AND HYPERBOLA. 


69 


R l ; and therefore,, by Prop. V. c l 2 : c f 2 : : r H x BOOK 
h l : d h 2 , and as c l is greater than c f, cl 2 is greater IL 
than cf 1 , and rh X h l is greater than d h 2 . To 
thefe add ch 2 , and (5. ii. and 47. i.) then c L 2 is greater 
than c d 2 . Confequently c l is greater than c D, and 
therefore r l is greater than k d. Again, by Prop. V. 
c l 2 : c f 2 : : d m 2 : G M X M f, and therefore, as above, 
d m 2 is greater than gm x m f. To thefe add the 
iquare of c M, and then {5. ii. and.47. i.) c d 2 is greater 
than c F 2 . Confequently k d is greater than f g. 

Part II. Let ab,l m be the axes of the oppolite hy- Fig. 56. 
perbolas ebd, af, and fd,g h any other conjugate 
diameters; then a b is lefs than the tranfverfe diameter 
F d, and l m is lefs than g h. 

For let d e be an ordinate to the axis a b, and let 
it meet it in 1, and let d t touch the hyperbola in d, 
and let it meet a b in t. Let b p touch the hyperbola 
in the vertex b, and let it meet the tangent d t in p, 
and let c be the center. Then, by Def. VII. c 1 d is 
a right angle, and therefore (19. i.) c d is greater than 
c 1, and confequently much greater than c b. Hence 
the tranfverfe axis is lefs than the tranfverfe diameter 
f d. Again, by Cor. 2. Prop. IV. g h is parallel to 
T d, and d 1 to p b ; and, by Prop. VII. c 1 : c b : : 

C b : c t, and therefore by converfion, c 1 : b 1 : 2 
c b : b t. Hence, as c 1 is greater than c b, (14. v.) 
b 1 is greater than b t. But (2. vi.) bi:bt::up:pt, 
and therefore d p is greater than p t ; and, as (29. i.) 
p b t is a right angle, pt (19. i.) is greater than b p. 

Hence d p is greater than b p ; and as, by Cor. 2. 

Prop. V. D p : B p : : c H : c L, c ii is greater (14. v.) 
than c l. Confequently the axis l m is lefs than the 
fecond diameter g h . 


F 3 


DEFI- 





OF THE ELLIPSE AND HYPERBOLA, 


BOOK 

II. 


Fig. 6o. 
6 1 . 


Fig. 6o. 


Fig. Ci. 


DEFINITIONS. 

X. 

In the ellipfe the greater axis is called the Tranfverfe 
Axis , and the other axis is called the Conjugate Axis ; 
and in the hyperbola the axis which is a tranfverfe dia¬ 
meter is called the Tranfverfe Axis, and the other axis 
is called the Conjugate Axis. 

XI. 

If c be the center* a b the tranfverfe axis, and d e 
the conjugate axis of the ellipfe a d b, or of the oppo- 
fite hyperbolas a i, b p, then if in ab two points f, o 
be fo taken that the rectangle under af, f b, and alfo 
the re&angle under a o, o b, be equal to the fquare of 
c d or c e, the femiconjugate axis; the points f, o are 
called the Foci, or Umbilici, of the ellipfe, hyperbola, 
or oppofite hyperbolas. 

Cor. i. As (axiom i. I.) af x fb is equal to A o X 
o b, the foci F, o are equally diftant from the vertices 
A, b, by the fixth Lemma. It is alfo evident, that the 
foci are equally diftant from the center. 

Cor. 2. In the ellipfe the diftance of each of the foci 
from either extremity of the conjugate axis is equal to 
the femitranfverfe axis. For, fuppofing a ftraight line 
to be drawn from d to o, the fquare of d o (47. i.) 
will be equal to the fquares of c o, c d together ; and 
therefore, by this Definition, (and 5. ii.) the fquare of 
d o is equal to the fquare of A c. Confequently d o 
is equal to ac; and therefore if with d or e as a cen¬ 
ter, and A c or c b as a diftance, a circle be defcribed, 
the circumference will cut A b in o, and f the foci. 

Cor. 3. In the hyperbola the diftance of each of the 
foci from the center is equal to the diftance between 
the vertices of the tranfverfe and conjugate axes. For, 
fuppofing a ftraight line to be drawn from d to A, the 
fquare of d a (47. i.) will be equal to the fquares of 

C A, 



OF THE ELLIPSE AND HYPERBOLA. 

c A, c d together; and therefore, by this Definition, 
(and 6 . ii.) the fquare of D A is equal to the fquare of 
c o or c F. Confequently c o or cf is equal to da; 
and therefore the foci f, o may be eafily found from 
the axes. 

Cor. 4. The double ordinate t s to the axis a b, 
drawn through either focus, fuppofe f, is equal to the 
parameter of the axis a b. For, by Prop. V. c b 2 : 
CD 2 :: AF x F B, or by this Def. as c d 2 : t f" ; and 
therefore (22. vi.) cb:cd:: cd:tf. Confequent- 
ly ( I 5 - 5 *) a b : o E *• : D e : t s, and the Cor. is evi¬ 
dent from Def. IX. 

XII, 

If through the foci f, o of an ellipfe a d b, or of the 
oppofite hyperbolas A 1, b p, ordinates f t, o i to the 
axis A b be drawn, and if through the points T, I in 
"which they meet the curve flraight lines h t, i l be 
drawn to touch the fection, or oppofite hyperbolas, the 
tangents ht,il are called Focal Tangents . 

PROP. XII. 

If a tangent , paffmg through a vertex of the tranfverfe 
axis of an ellipfe or oppofte hyperbolas , meet a focal tan- 
gent , its fegment between the point of contaft and point 
of concourfe will be equal to the fegment of the axis be¬ 
tween the point of contaft and the focus , to which the 
focal tangent belongs. 

Let t b be an ellipfe or hyperbola, of which A b is 
the tranfverfe axis, and f, o the foci, and let a h touch 
the ellipfe, or either of the oppofite hyperbolas, in the 
vertex a, and meet in the point h the focal tangent 
h g, belonging to the focus f ; the fegment a h is 
equal to the fegment a f. 

For let h g touch the fe&ion in t, and t f, being 
E 4 drawn. 




BOOK 

II. 


Fig. 60. 

6i. 


Fig. 

61. 



7 * 


BOOK 

II. 


Fig. 60. 
4n. 


OF THE ELLIPSE AND HYPERBOLA, 

drawn, will be an ordinate to a b, by the twelfth 
Definition. Let b g touch the fe&ion in the vertex b, 
and meet h g in g ; and let c be the center, and d e 
the conjugate axis. Then, by Cor. 2. Prop. IV. a h, 
d e, t f, b g are parallel; and therefore, by Cor. to 
Prop. XIII. Book I. A h : b g : : h t ; T G. But it 
is evident, (from 10. vi.) that ht:tg::af:fbj 
and therefore (11. v.) ah : e g : : a f : f b. Confe- 
quently a h x b g is limiiar to af X f b, and, by 
Prop. IX. andDef. XI. each of thefe re&angies is equal 
to the Iquare of c d. They are therefore equal to one 
another; and, as they are alfo fimilar, a h is equal to 
A f, and B g is equal to b f. 

Cor. If o 1 be drawn an ordinate to a b, and on the 
lide of a b oppofite to that on which f t is, and if 
through 1, the point in which it meets the curve, there 
be drawn the focal tangent k l, meeting the tangent 
H a in k, and the tangent gb in l ; then h l will be 
a parallelogram, and each of the oppofite tides h k, 
g l will be equal to the tranfverfe axis A b. For, by 
the above, and Cor. 1. to Def. XI. a k, a o, b f, b g 
are equal to one another, and alfo ah,af,bo,bl to 
one another. Confequently h k, g l are equal and 
parallel, and therefore (33.1.) h g, k l are equal and 
parallel. Hence h l is a parallelogram ; and as a h 
is equal to a f, and A k to b f, h k or g l is equal to 
A B. 

PROP. XIII. 

If from any point in the curve of an ellipfe or hyperbola 
two Jlraight lines he drawn to the foci , their fum in the 
ellipfe , hut their difference in the hyperhola > will he equal 
to the tranfverfe axis . 

Let p be any. point in the curve of the ellipfe, or 
hyperbola ptb, of which a b is the tranfverfe axis, 

and 



OF THE ELLIPSE AND HYPERBOLA* 

and the points f, o the foci, and let p o, p f, be 
flraight lines drawn to the foci; the fum of po 3 pf 
in the ellipfe, but their difference in the hyperbola, is 
equal to a b. 

For, the reft remaining as in the preceding Propo- 
fition and its Corollary, let p r be drawn an ordinate 
to a b, and let it meet the curve again in m, the focal 
tangent g h in n, and the focal tangent k l in a. 
Then, by Cor. s. Prop. IV. h k, o i, d e, n q, t f, 
g l are parallel, and therefore, by Prop. XIII. Book I. 
T H 2 : t n 2 : : A H 2 : m n X N p. But, on account of 
the parallels (io.-vi.) t h 2 : t n 2 : : f a 2 : f r 2 ; and 
therefore (i i. v.) f a 2 : f r 2 : : a h 2 : m n X n p, and 
as, by Prop. XII. f a is equal to a h, f a 2 is equal to 
a pi 2 , and therefore f r 2 is equal to m n X n p. To 
thefe equals add the fquare of p r, and then (6. ii. and 
47. i.) the fquare of n r is equal to the fquare of r f, 
and confequently n r is equal to r f. Again, by Prop. 
XIII. Book I. 1 l 2 : 1 a 2 : : b l 2 : p a X a m ; and 
on account of the parallels (jo. vi.) 1 l 2 : 1 a 2 : : o b 2 : 
o r 2 , and therefore (11. v.) o b 2 : o r 2 : : b L 2 : p a X 
a m. But, by Prop. XII. o b is equal to b l, and 
therefore o b 2 is equal to b l 2 , and (14. v.) o r 2 is 
equal to p a x a m. To thefe equals add the fquare 
of r m, or its equal the fquare of r p, and then (6. ii. 
and 47. i.) the fquare of o m in the ellipfe, and the 
fquare of o p in the hyperbola, is equal to the fquare 
of r a. Confequently in the hyperbola o p is equal 
to r q, and in the ellipfe o m is equal to r q. But in 
the ellipfe pr, r m are equal, and the angles o r m, 
o rp are equal, being right angles, and o r is common 
'to the two triangles o r m, o r p, and therefore (4. i.) 
o m is equal to o p. In each fe&ion therefore o p is 
equal to r q, and p f to n r. Confequently in the el¬ 
lipfe the fum of p o, p f, but in the hyperbola their 

dif~ 


73 

BOOK 

II. 



74 


OF THE ELLIPSE AND HYPERBOLA. 


BOOK 

II. 


feff- 57- 

58. 


difference is equal to n q. But a h is a parallelogram, 
and therefore (34*. i.) no, h k are equal; and as, by 
the Cor. to Prop. XII. h k is equal to A b, the fum of 
p o, p f in the ellipfe, but their difference in the hy¬ 
perbola is equal to a b, the tranfverle axis. 

Cor. 1. If from any point in the curve of an ellipfe, 
or hyperbola, two flraight lines be drawn to the foci, 
in the ellipfe the difference between the tranfverfe axis 
and either of the two will be equal to the other ; but 
in the hyperbola the fum of the tranfverfe axis and the 
leaft of the two will be equal to the other. 

Cor. 2. If the conjugate axis de be produced till it 
meet the oppofite focal tangents in v and w, each of 
the fegments c v, c w between c the center and a fo¬ 
cal tangent will be equal to the fe mi tranfverfe axis. 
For, let the oppofite focal tangents meet the tranfverfe 
axis a b in x and y. Then, as, by Cor. 1. Def. XI. c f, 
c o are equal, it is evident, from Prop. VII. that c y, 
c x are equal; and as x w, v y are parallel, the angles 
(29. i.) c x w, c y v are equal. Confequently, as the 
angles at c are right angles, (26. i.) c v is equal to 
c w. In each fection therefore the Cor. is evident. 

PROP. XIV. 

If from a point without an ellipfe or oppofte hyperbolas 
two flraight lines he drawn to the foci , their fum in the 
ellipfe will be greater , but their difference in the hyper¬ 
bola will be lefs , than the tranfverfe axis. But if front 
a point within an ellipfe or hyperbola two flraight lines 
be drawn to the foci , their fum in the ellipfe will be lefs , 
but their difference in the hyperbola greater , than the 
tranfverfe axis. 

Part I. Let e be a point without the ellipfe A d e, or 
oppofite hyperbolas a, b, and let ef,eo be ftraight 

lines 



OF THE ELLIPSE AND HYPERBOLA. 7 1 

lines drawn to the foci f, o ; the fum of e f, e o in book 
the ellipfe is greater, but their difference in the hyper- IL 
bolas lefs, than a b the tranfverfe axis. 

In the ellipfe let e f cut the curve in d, and draw Fig. 57. 
o D ; and then (20/i.) oe,ed together being greater 
than o d, the three o e, e d, d f together are greater 
than o d, n f together. Confequently, by Prop. XIII. 
the fum of ef, e o is greater than a b. In the hy¬ 
perbolas let e f be greater than e o, and let e o cut Fig. 5S, 
the curve of the hyperbola a in d, and draw fd. Then 
d f, d e together are greater than e f. But, by the 
Cor. to Prop. XIII. d f is equal to a b, o d together, 
and therefore ab,od, d e together, or a b and o e 
together, are greater than e f. Confequently the dif¬ 
ference between ef, eo is lefs than a b. 

Part II. Let g be a point within the ellipfe or hy¬ 
perbola A D, and let the ftraight lines g f, go be 
drawn to the foci f, o ; the fum of g f, g o in the el¬ 
lipfe is lefs, but in the hyperbola their difference k 
greater, than a b the tranfverfe axis. 

In either fe&ion let g f meet the curve in d, and 
draw d o. Then, in the ellipfe, o d, d g together 
(20. i.) are greater than o g; and therefore o d, d g, 
g f together, or o d, d f together, are greater than o g, 
g f together. Confequently the fum of gf, go is 
lefs than a b, by Prop. XIII. In the hyperbola o d, 
d g together (20. i.) are greater than g o, and there¬ 
fore o d, D G, a b together are greater than g o, a b 
tpgether. But, by Cor. t. to Prop. XIII. o d, a b toge¬ 
ther are equal to d f, and therefore d f, d g together, 
or f G, are greater than g o, a b together. Confe¬ 
quently the difference between f g, g o is greater than 
A B. 

Cor. 1. From this and Prop. XIII. it is evident, that 
two ftraight lines being drawn from a point to the foci 

of 



7 * 

BOOK 

II. 


F’g- 59 - 


Tig. 6 :. 
63 - 




OF THE ELLIPSE AND HYPERBOLA# 

of an cllipfc or hyperbola, if in the ellipfe their fum be 
greater, or in the hyperbola their difference be lefs, 
than the tranfverfe axis, the point will be without the 
fettion. If in the ellipfe the fum of the two lines, or 
in the hyperbola their difference, be equal to the tranf¬ 
verfe axis, the point will be in the curve of the fe&ion. 
Laffly, if in the ellipfe the fum of the two lines be lefs, 
or in the hyperbola their difference be greater, than 
the tranfverfe axis, the point will be within the fedtion. 

Cor . 2. If o, f be the foci of the hyperbola b p, and ; 
if the fide o d, of the triangle o d f, be equal to A b, 
the tranfverfe axis, and o d f be an obtufe angle, then 
the ftraight line o d produced will meet the curve of 
the hyperbola b p, in which the focus f is fituated. 
For let o d be produced to k, and make the angle dfl 
equal to the angle f'd k. Then, as by hypothefis 
o d f is an obtufe angle, k d f is an acute angle, and 
therefore, as the angle d f l is equal to it, the ftraight 
lines d k, f l, being produced, will meet. Let them 
meet in p, and (6. i.) v f will be equal to pd. Con- 
fequently, as the difference of po,pf is equal to o d, 
or a b, the point p is in the curve of the hyperbola, by 
the preceding Corollary. 

PROP. XV. 

If from any point in the curve of an ellipfe , or hyperbola,\ 
two ftraight Vines be drawn to thefoci, the ftraight line 
.bifefting the angle adjacent to that contained by them 
will touch the ellipfe ; but We ftraight line bifetting the 
angle contained by them will touch the hyperbola . 

From the point p, in the curve of the ellipfe or hy¬ 
perbola 1 b p, let two ftraight lines p f, p o, be drawn 
to the foci F. o, and in the ellipfe let o p be produced 
to d ; the ftraight line p e bifecling the angle fpd, 

ad- 






J Basire sc. 





























































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OF THE ELLIPSE AXD HYPERBOLA. 77 

adjacent to fpo in the ellipfe, touches the ellipfe; book 
but the ftraight line p e, b‘ife£ting the angle f p o in 1L 
the hyperbola, will touch the hyperbola. 

In the ellipfe let p d, the part, of o p produced, be- 
equal to f f. Draw f d, and let it meet p e in e. 

Then, as f p, p d are equal, and p e common to the 
two triangles p e f, p e d, and the angle f p e equal 
to the angle d f e, the fide f e (4. i.) is equal to d e, 
and the angles f e p, d e p are equal. In ep take any 
point g, and draw g o, g f, g d. Then, as f e is 
equal to e d, and as the angles f e g, d e g are equal, 
we have f g (4. i.) equal to d g. But (20. i.) d g, 
g o together are greater than d o, or o p, p f toge¬ 
ther ; and therefore, by Prop. XIII. d g, g o together 
are greater than A b the tranfverle axis. Confequently 
go, g f together are greater than a b, and therefore, 
by Cor. 1. to Prop. XIV. the point g is without the 
ellipfe b p, and confequently p e touches it in p. 

In the hyperbola take p d in p o equal to p f. Draw Fig. C3. 
r d, and let it meet p e in e. Then, as p f, p d are 
equal, and as p e is common to the two triangles fpe, 
d p e, and as the angles f p e, d p e are equal, the 
fide f e (4. i.) is equal to the fide e d, and the angles 
f E p, d e p are alfo equal. I11 e p take any point g, 
and draw go, g d, g f. Then, as f e, e d are equal, 
and as the angles f e g, d e g are alfo equal, and e g 
common to the two triangles f e g, d e g, the fide 
f g (4. i.) is equal to the fide d g. Alfo, as p d is 
equal to p f, by Prop. XIII. d o is equal to A b, the 
trail!verfe axis. But g d, d o together (20. i.) are 
greater than g o, and therefore g f and a e, together 
are greater than g o. Confequently the difference be¬ 
tween g 0 and g f is lefs than a b, and therefore, by 
Cor. I. to Prop. XIV. the point g is without the hy¬ 
perbola, and p e touches the hyperbola. 

Cor . 



73 


OP THE ELLIPSE AND HYPERBOLA# 


BOOK 

II. 


Fig. 62. 


Fig. 

63 - 


Fig. 64. 
65 - 


Fig. 66. 

67 . 




Cor. 1. From this Prop, and Prop. VI. Book I. it is 
evident, that if a ftraight line touch an ellipfe or by- I 
perbola, and ftraight lines be drawn from the point of 
conta6l to the foci, in the ellipfe the tangent will bife£l 
the angle adjacent to that contained by thefe two 
ftraight lines drawn to the foci; but in the hyperbola 
the tangent will bife£t the angle contained by thefe 
two ftraight lines drawn to the foci; In the ellipfe 
the angle o p g (15. i.) is equal to the angle f p e. 

Cor. 2. If from the foci o, f of an ellipfe, or hyper- H 
bola, two ftraight lines, o d, f d be drawn to a third 
point d, of which o d, one of them, is equal to the 
tranfverfe axis a b, and if the other fd be bife&ed • 
in e, by a ftraight line p e at right angles ; the per- ■ 
pendicular p e will fomewhere touch the fe&ion, pro- \ 
vided, in the hyperbola, o d f be an obtufe angle, j 
And, on the contrary, if p e touch the fe&ion and bi~] 
fe£t f d in e at right angles, then o d will be equal 
to the tranfverfe axis. This is evident from Cor. 2. | 
Prop. XIV. Prop. VI. Book I. and the above demon - 
ftration. 

Cor. 3. The reft remaining as above, let f g be at : 
right angles to the ftraight line l g, touching the el¬ 
lipfe or hyperbola in l, and let f g be produced to h, 
fo that g h may be equal to f g ; then a ftraight line, { 
bife&ing d h at right angles, will pafs through the fo- ; 
cus o. For, by the preceding Cor. o h is equal to the 
tranfverfe axis, and conlequently equal to o d. If there- , 
fore o k be drawn, bife&ing d h in k, the angles (8. i.) 
o k d, o k h will be equal. Hence the Cor. is evident. 

Cor. 4. The reft remaining as in the demonftration of 
the Propofition, let d f be fo divided in l, that d l may 
be to l f as a b to o f, or, which is the fame thing, as 
d o to o f, and in d f, produced as in the figures, let d n 
be to n f as a b to o f, and then a circle deferibed upon 

L N 




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OF THE ELLIPSE AND HYPERBOLA. 

i. N its a diameter will pafs through the focus o. For 
in the ellipfe produce d o to m, but in the hyperbola 
produce f o to m, and draw o l, o n. Then the an¬ 
gles dof, F o m * (3, vi.) in the ellipfe are bife&ed by 
the ftraight lines o l, o n ; and as the angles d o f, 
f o m together {13. i.) are equal to two right angles, 
the angles lof,fon together are equal to a right 
angle. But in the hyperbola the angles dof,.dom* 
are bife£led (3. vi.) by the ftraight lines o l, on; and 
as the angles d o f, d o m together are equal to two 
right angles, the angles t, o d, don together are 
equal to a right angle. In either cafe therefore the an¬ 
gle l o n is a right angle, and confequently (31. iii. 
and 21. i.) a circle deferibed about l n as a diameter 
will pafs through o. 

Sir Ifaac Newton makes much ufe of the properties 
expreffed in the three lad Corollaries. See the Prin¬ 
cipal, Seed. IV. Book L 

PROP. XVI. 

If a firelight line touching an ellipfe or hyperbola meet a 
Jlraigbt line drawn from either of the foci, and be at 
right angles to it } the Jlraigbt line joining the center and 
the point of concourfe will be equal to the femitranfverfs 
axis : or , if a Jlraigbt line touch an ellipfe or hyperbola , 
and Jlraigbt lines be drawn from the point of contact to 
the foci, a fl raight line drawn from the center to the 
tangent , and parallel to either of the two drawn to the 
foci , will he equal to thefemitranfverje axis , 

Part I. Let the ftraight line g p, touching the ellipfe 
or hyperbola p b in the point p, meet in the points g, 

* That the angle fom in the ellipfe, and the angle D o m in the 
hyperbola, is bife&ed by on, is proved by Sindon, in his Prop. A. in 
the fijctli Book of his edition of Euclid- 


79 


BOOK. 

II. 


Fig. 


e, the 


8 o 


OP THE ELLIPSE AND HYPERBOLA. 


BOOK 

II. 


Fig. 68. 
69 . 


E, the ftraight lines f g, o e, drawn from the foci f, o, 
and be at right angles to them, and from c the center 
draw c g, c e to the points of concourfe ; each of the 
ftraight lines c g, c e is equal to A c or c b, the femi- 
tranfverfe axis. 

For, draw f p, op, and let f p, or fp produced, 
meet o e produced in d. Then, as p e o is a right 
angle, p e d is a right angle; and as pe is common 
to the two triangles peo, p e d, and as, by Cor. 1. 
to Prop. XV. the angles ope,dpe are equal, the 
fide p d (26. i.) is equal to the fide p o, and d e is 
equal to e o. Confequently, by Prop. XIII. f d is 
equal to the tranfverfe axis a b ; and as f c, c o are ! 

equal, and d e equal to e o, f o : c o : : d o : e o. . 

The ftraight lines f d, (2. vi.) c e are therefore paral¬ 
lel, and o f : f D : : o c : c e. But o c is the half of 
o f, and therefore c e is equal to the half of fd. Con- ] 
fequently c e is equal to a c or c b, the femitranfverfe 
axis. If f g, p o be produced till they meet in h, it 
may be proved, in the fame manner, that f g is equal 

to g h, f p to p h, h o to A b, and c g to A c or c B. 

Part II. Let the ftraight line g p touch the ellipfe or 
hyperbola p b in the point p, and let p f, p o be 
ftraight lines drawn from the point of con tad to the 
foci f, o. Let c be the center, and draw c e parallel 
to f p, and c g parallel to o p, and let c e meet the 
tangent in e, and c g meet it in g 5 each of the 
ftraight lines c e, c g is equal to a c or c b, the fcmi- 
tranlverfe axis. 

For draw o e, and, being produced, let it meet f p, 
or f p produced, in d. Then, as f d, c e are parallel, 

(2. vi.) f c : c o : : d e : e o, and as, by Cor. to 
Def. XI. f c, c o are equal, d e is equal to e o. By 
Cor. 1. to Prop. XV. the angles ope, d p e are equal, 
and therefore (3. vi.) Oe;ed::oP:pd, and as 

O E, 




OF THE ELLIPSE AND HYPERBOLA. 


8l 


o e, e d are equal, o p is equal to pd. Confequently, BOOK 
by Prop. XIII. f d is equal to a B, the tranfverfe axis ; 1L 
and as f d, c e are parallel, fo :fd::co:ce. 

But c o is the half of f o, and therefore c e is the half 
of f d. Confequently c e is equal to a c or c b, the 
femitranfverfe axis. If f g be drawn, and, being pro¬ 
duced, meet o p produced in h, it may be proved, in 
the fame manner, that f g is equal to g h, f p to r h, 
and c g to a c or c b. 

Cor. The reft remaining as above, if the ftraight line 
i k, drawn through c the center, and parallel to the 
tangent g p, meet p f in i and p o in k, the fegments 
v i, p k are equal ; and each of them is equal to A c 
or c b. For (34. i.) p k is equal to c g and p 1 is equal 
to c E. 

The demonftrations of the nth and 12th Propoft- 
tions of the firft Book of the Principia depend, in a 
very confiderable degree, on this property. 

PROP. XVII. 

The reft angle contained under two Jlraight lines , drawn 
from the foci of an ellipfe or hyperbola to a tangent , and 
at right angles to it , is equal to the fquare of the femi - 
conjugate axis. And the rectangle contained under two 
jlraight lines , drawn from the tranfverfe axis of an el¬ 
lipfe or hyperbola to a tangent , and at right angles to 
it , is equal to the fquare of the femiconjugate axis, if one 
of them be drawn from the center , and the other meet 
the tangent in the point of conta&. 

Part I. Let the ftraight lines f g, o e, drawn from Fig. 70. 
the foci f, o of the ellipfe or hyperbola p B, meet in 
the points g, e, the ftraight line g e which touches 
the feeftion in p, and let them be at right angles to the 
tangent g e, and c being the center, let c d be the fe- 
G miconjugate 



82 


BOOK 

II. 


OF THE ELLIPSE AND HYPERBOLA. 

miconjugate axis ; the re&angle under f g, o e ia* 
equal to the fquare of c d . 

Draw e c, and, being produced, let it meet f g, or 
f g produced, in H. Then as f g, o e are at right 
angles to gpe, they are parallel to one another, (28.i.) 
and (29. i.) the angles c f h, c o e are equal. The 
triangles (15. and 32. i.) c f h, c o e are therefore 
equiangular, and, Cor. 1. to Def. XI. c f is equal to 
c o. Confequently (2 6 . i.) c h is equal to c e, and 
f h is equal too e. If therefore, with c as a center, 
and c a or c e as a difiance, a circle be defcribed, it 
will pafs through the points e, g, by Prop. XVI. and 
confequently through h ; and therefore (35. and 36. iii.) 
the re&angle under g f, f h is equal to the re&angle 
under af, f b. But as f h is equal to o e, the rect¬ 
angle under gf, e o is equal to the rectangle under 
a f, f b ; and as, by the eleventh Definition, the reCt- 
angle under a f, f b is equal to the fquare of c r>, the 
reCtangle under g f, e o is alfo equal to the fquare of 

G D. 

Part II. Let a e be the tranfverfe axis of the ellipfe- 
or hyperbola p b, of which c is the center, and let G f , 
touch the feCtion in the point p; the reCtangle under 
the flraight lines c k, m p, drawn from the tranfverfe 
axis a b to the tangent g e, and at right angles to it, 
is equal to the fquare of c d, the femiconjugate axis. 

Let the conjugate axis meet the tangent in the point 
1 ; and from the point p draw pn an ordinate to a b, 
and p l an ordinate to d c 1. Then, as c k, m p are at 
right angles to the tangent g e, they are (28. i.) pa¬ 
rallel to one another, and, by Cor. 2. Prop. IV. as p n 
is an ordinate to a b it is parallel to d c 1, and as p l is 
an ordinate to D c 1 it is parallel to a b. Confequently 
(29. i.) the angle k c b is equal to the angle p m n ; 
and as 1 c b, p n m are right angles, the angles 1 c k, 

K C B 



OF THE ELLIPSE AND HYPERBOLA. 

kcb together are equal to the angles m p n, p m n 
together, and therefore the angles ick, m p n are 
equal,.and the triangles ick,mpn are equiangular. 
Hence ck:ci::pn:pm. But (34. i.) p n is equal 
to c l, and therefore ck:ci::cl:pm, and (16. vi.) 
c k X p m is equal to c 1 x c l. Confequently, as 
the rectangle under c 1, cl, by Prop. VII. (and 17. vi.) 
is equal to the fquare of c d, the rectangle under c k, 
p m is alfo equal to the fquare of c d. 

PROP. XVIII. 

If a freight line touching an ellipfe or hyperbola be li¬ 
mited by tangents paffzng through the vertices of the 
tranfuerfe axis , the circumference of a circle deferibed 
about it as a diameter will pafs through the foci ; and 
the rectangle under the two firaight lines drawn from 
the point in which it touches the Je biion to the foci will 
he equal to the fquare of the femidiameter parallel to it. 

Part I. Let the ftraight line g e touch the ellipfe or 
hyperbola p b in the point p, and meet in the points 
c, e the tangents a g, b e, palling through the ver¬ 
tices A, b of the tranfverfe axis a b ; the circumference 
of a circle deferibed about g e as a diameter will pafs 
through the foci f, o. 

For, by the feventh Definition, and Cor. 2. Prop. IV. 
gab, e b a are right angles; and, by the eleventh 
Definition, and Prop. IX. the rectangle under a g, 
b e is equal to the rectangle under a o, ob. Con- 
fequently (17. vi.) e b : b o : : a o : a g, and there¬ 
fore the ftraight lines e o, g o being drawn, the angle 
E o b ( 6 . vi.) is equal to the angle A G o, and the 
angle b e o is equal to the angle aog. The angles 
e o b, a o g together are therefore equal to a right 
angle, and confequently (32, i>) the angle goe is a 
g 2 right 


*3 


BOOK 

II. 


Fig. 12. 
73 » 



s 4 


BOOK 

II. 


rig. 70. 
7 *. 


OF THE ELLIPSE AND HYPERBOLA. 

right angle. If therefore a circle be defcribed about 
G e as a diameter, it is evident (from Prop. 31. iii. 
and 31. i.) that its circumference muft pafs through 
the focus o; and in the fame way it may be proved, 
that it muft pafs through the focus f. 

Part II. The reft remaining as above, let p o, p f be 
drawn to the foci o, f, and let c d be the femidiameter 
parallel to ge; the re&angle under p f, p o is equal 
to the fquare of c d. 

For, draw o 1 perpendicular to g e, and, being pro¬ 
duced, let it meet p f, or p f produced, in h. Then, 
by Cor. 1. to Prop. XV. the angles o p i, h p i are 
equal, and the angle o 1 p is equal to the angle hip, 
each of them being a right angle, and p 1 is common 
to the two triangles opi, hpi. Confequently (26. i.) 
p o is equal to p h, and o 1 to 1 H ; and it is evident 
(from 3. iii.) that the point h muft be in the circum¬ 
ference of the circle defcribed about g e as a diameter, 
and palling through f, o, according to Part I. The 
re&angle under f p, p h, or that under f p, p o, (35. 
and 36. iii.) is therefore equal to the rectangle under 
g p, pe. Confequently, by Prop. IX. the redtangle 
under f p, p o is equal to the fquare of c d. 

PROP. XIX. 

A flraight line drawn from either of the foci of an ellipfc 
or hyperbola , perpendicular to a tangent , is to a Jlraight 
line drawn from the fame focus to the point of conta&i 
as the femiconjugate axis to the femidiameter parallel to 
the tangent . 

Let p e be an ellipfe or hyperbola, of which the foci 
are f, o, and let g e touch the fe&ion in p, and let 
f g be perpendicular to it; the perpendicular f g is to 
tlie ftraight line f p, joining the focus f and point of 

con- 



OF THE ELLIPSE AND HYPERBOLA. 

contaft, as c d the femiconjugate axis to c r the femi- 
diameter parallel to g e. 

For, draw o e perpendicular to G e, and draw p o. 
Then, by Cor. i. to Prop. XV. the angles fpg, ope 
are equal; and f g p, o e p being right angles, the tri¬ 
angles f p G, ope, are equiangular. Confequently 
(4. vi.) f g : F p : : o e : o p, and by alternation f g : 
oe::fp:op; and therefore (22. vi.) F G x o e : 
F p X o p : : f g 2, : F p 2 . But, by Prop. XVII. F G x 
o E is equal to c d 2 , and F p x o p is equal to c r 2 , by 
Prop. XVIII. Confequently c d 2 : c r 2 : : f g 2 : F p 2 , 
and therefore (22. vi.JcD : c r : : fg : fp. 

Cor. t . The reft remaining as above, let c E parallel 
to f p meet the tangent g e in e, and let c k be per¬ 
pendicular to g e and meet it in k. Then (4. vi.) 
F G : f p : : c k : c e. But, by Prop. XVI. c e is 
equal to c b, the femitranfverfe axis, and therefore, by 
the above, (and 11. v.) ck:cb::cd:ck. 

Cor . 2. The reft remaining as above, let the ftraight 
line p m, perpendicular to the tangent ge, meet the 
tranfverfe axis a b in m, and then c b will be to c d as 
c r to p m. For, by the preceding Cor. ck:cb:: 
c d : c r, and therefore (1. vi.) ckxpm:cbx 
p m : : c d 2 : c R X c d. But, by Prop. XVII. c k x 
p m is equal to c d 2 , and therefore (14. v.) cbxpm is 
equal tocRXco. Confequently (16. vi.) cb : c d : : 
c r : p M. 

PROP. XX. 

If ajlrdigbt line touch an ellipfe or hyperbola , and from 
the point of contact two jlraigbt lines be drawn to an 
axis , the one an ordinate to it, the other perpendicular to 
the tangent , the Jegment of the axis between the center 
and ordinate will be to the fegment between the per¬ 
pendicular and ordinate as the axis to its parameter. 

. G 3 Let 


8 5 ‘ 

BOOK 

II. 



86 


OF THE ELLIPSE AND HYPERBOLA. 


BOOK 

II. 


Fig- 74- 
75- 


Let the ftraight line i r touch the ellipfe or hyper¬ 
bola p b in the point p; and a b being the tranfverfe, 
and d E the conjugate axis, and c the center, let p h 
be an ordinate to A b, and pg an ordinate to d e ; and 
let the ftraight line m p be perpendicular to i r, and 
meet a b in k and d e in m ; then c h is to k h as 
a b to its parameter, and c g is to g m as de to its 
parameter. 

Let the tangent i r meet a b in r, and d e in i. 
Then, by Cor. 2. Prop. VII. the redtangle under c h, 
h r is equal to the redtangle under A h, h b ; and 
r p k being a right angle, and p h being at right an¬ 
gles to k r, the redtangle under k h, h r is equal to 
thefquare of ph (Cor. 8. vi. and 16. vi.). But (r. vi.) 
ch:kh::chXhr:khxhr; and therefore, 
on account of the equals, (and 11. v.) ch : kh 
A h x h b : p h\ Confequently, as by Prop. VI. 
ah x h b is to p h* as a b to its parameter, c h is to 
K h (11. v.) as A b to its parameter. 

Again (1. vi.) in the ellipfe, cg: gm: : c g X g 1 ; 
G M X G I ; and as above, by Cor. 2. Prop. VII. the 
rectangle under c g, g i is equal to the rediangle un¬ 
der e g, G d, and (Cor. 8. vi.) the redtangle under 
G m, g 1 is equal to the fquare of p g. Confequently 
c G : G M : : e g X g d : p g 2 ; and therefore, by 
Prop. VI. (and 11. v.) .c G is to g >1 as d e is to its 
parameter. 

Laftly, in the hyperbola, as, by Cor. 2. Prop. IV. 
p g is parallel to A b, and p h parallel to d e, c g is to 
g m as k p to p m, and therefore as k h to h c. But 
(1. vi.) kh:hc::khxrh ; h c x r h ; and 
(Cor. 8. vi. and 17. vi.) k h x r h is equal to p h% 
and, by Cor. 2. Prop. VII. h c x r h is equal to a h 
X h b. Confequently ( n. v.) c g : g m : : P h 2 : 
A h x h B. But, by Def. VIII. p n 2 is to a h x h b 

as 



PlateXI .page file 



J. Bum re >>v 



































OF THE ELLIPSE AND HYPERBOLA. 

as dcHocb 2 ; and d c 2 is to c B 2 as d e 2 to ab 1 ; or, 
as is evident from Def. IX. d e 2 is to a b 1 as de is to 
its parameter. Confequently c G is to G m as D E is to 
its parameter. 


PROP. XXL 

Ij a Jlraight line touch an ellipfe or hyperbola, and a 
Jlraight line be drawn from the point of contact at right 
angles to it, and meet the axes , the rectangle under the 
fegmenis of the perpendicular, between the point of con - 
taEl and the axes , will be equal to the fquarc of the J'e- 
nu diameter parallel to the tangent . 

Let the ftraight line i r touch the ellipfe or hyper¬ 
bola p r in the point p, and let the ftraight line m p, 
at right angles to i r, meet the tranfverfe axis A b in 
k, and the conjugate axis d e in m, and, c being the 
center, let c l be the femidiameter parallel to i 11; the 
rectangle under pm, p k is equal to the fquare of c l. 

For, by Cor. 2 . Prop. XIX. and inverfion, p k : c l : : 
c d : c b, and therefore P k 2 : cl 2 : : c D 2 : c b 2 . 
And p g being drawn an ordinate to d e, by Prop. XX. 
c g is to g m as d e to its parameter. But, by Def. IX. 
(and Cor. 2. to 20. vi.) d e is to its parameter as d e 2 
to a b 2 , or (15. v.) as c d 2 to c b 2 ; and as p g is paral¬ 
lel to A b , (2. vi.) c G : G M i : p K : P m. Conse¬ 
quently (11. v.) p k : p m : : CD 2 : c B 2 ; and there¬ 
fore (r. vi.) p k 2 : p m x p k : : c d 2 : c b 2 . But, by 
the above, p k 2 : c l 2 : : c d 2 : cb 1 ; and therefore 
(11. v.) p k 2 : P M x P K : : p K 2 : c l 2 . Confequently 
(14. v.) p m X p k is equal to c l 2 . 

Cor. By the above, and Prop. XVIII. the re&angle 
under p m, p k is equal to the re&angle under po, pf, 
the ftraight lines drawn from p the point of contact to 
the foci o, f. 

&4 


87 


BOOK 

II. 


Fig. 

7S- 


PROP, 



zb 


BOOK 

II. 


Fig. 76. 


OP THE ELLIPSE AND HYPERBOLA. 

PROP. XXII. 

If a circle be defcribed about the tranfverfe axis of an el- 
lipfe as a diameter , a polygon may be inf crib cd in it, and 
a correfponding polygon in the eUipfe^fo that the polygon 
in the circle Jhall be to that in the ellipfe as the tranf¬ 
verfe axis to the conjugate axis . 

Let a b be the tranfverfe, and f g the conjugate axis 
of the ellipfe afbg, and about a b as a diameter let 
the circle a d b e be defcribed ; a polygon may be in- 
fcribed in A d b e, and a correfponding one in the el¬ 
lipfe a f b g, fo that the polygon in the circle {hall be 
to that in the ellipfe as a b to f g. 

For, let c be the common center of the ellipfe and 
circle, and produce f g till it meet the circumference 
of the circle in d, e. Let k, m be points in the cir¬ 
cumference, and draw d k, km, m a. Draw k p, 
m o parallel to d c, and let them meet a c in p, o, and 
the curve of the ellipfe in l, n, and draw fl,ln,na. 
Draw k h, li parallel to a b, and let them meet d c 
in h, i. Then pi, ph are parallelograms, and (34. i.) 
c h is equal to p k, and c 1 equal to p l; and k p, 
m o are perpendicular to a c, and l p, n o are ordi¬ 
nates to a b, by Cor. 2. Prop. IV. By Prop. V. A c 2 : 
c f 2 : : a p x p B : pl 2 , and therefore (3^. iii.) as 
A p 'X p b is equal to k p 2 , a c 2 : c f 2 : : k p 2 : p l 2 , 
and (22. vi.) a c or d c : c F : : k p or c h : p l or 
c 1 *. Confequently (19. v.)dc:fc::dh:fij 
and dh:fi::ch:ci. But (1. vi.) c h : c 1 : : 
parallelogram p h : parallelogram p 1; and dh:fi;; 
the triangle d k h : the triangle fli, Confequently 

* This is the property referred to by writers on the Orthographical 
Projection of the Sphere, when they prove, that a circle, not parallel 
to the p’ane of projection, is projected into an ellipfe. 


(II. 



OF THE ELLIPSE AND HYPERBOLA* £9 

(n. and 12. v.) d c : f c : : the trapezium dkpc: BOOK 
the trapezium flpc, In the fame manner it may be 1L 
demonftrated that DC : fc : : the trapezium k m o p : 
the trapezium l n o p ; and alfo that d c : f c : : the 
triangle mao: the triangle n a o. But (15. v.) d c : 

F c : : a b : f g 3 and therefore (12. v.) a b : f g : : the 
polygon d k m a c : the polygon flnac, Confe- 
quently, as infcriptions may be made in a fimilar man¬ 
ner all round the circle and ellipfe, the Prop, is evident. 

Cor . A polygon may be infcribed in an ellipfe, which 
{hall be deficient from the ellipfe by a fuperficies lefs 
than any given fuperficies. For, the reft remaining as 
above, if a ftraight line parallel to l f be drawn to 
touch the ellipfe, and meet c f and p l produced, and 
from f, l ftraight lines be drawn to the point of con¬ 
tact, the triangle thus formed will be equal to half the 
parallelogram contained by l f, the tangent parallel 
to it, and c f, p l produced. This triangle therefore 
will be greater than half the elliptic fegment contained 
under the curve l f, and the ftraight line l f. Such, 
an infcription therefore being made all round the el¬ 
lipfe, the Cor is evident (from 1. x.). 

1 

PROP. XXIII. 

If the tranfverfe axis of an ellipfe be alfo a diameter of a 
circle , the ellipfe will be to the circle as the conjugate 
axis to the tranfverfe axis . 

Let A B be the tranfverfe axis of the ellipfe afbg, 
and alfo a diameter of the circle adbe; the ellipfe is 
to the circle as f g the conjugate axis to ab the tranf¬ 
verfe axis. 

For, every thing remaining as in Prop. XXII. let the Fig. ‘jS. 
circle a r s be to the circle adbe as f g to a b ; 
and then, if it can be proved that the circle q r s is 

equal 



90 


OF THE ELLIPSE AND HYPERBOLA., 


BOOK 

II. 


equal to the ellipfe afbg, the truth of the Proportion 
will be manifeft. If the circle qrs be not equal to 
the ellipfe, let it f rft, if poffible, be greater. Then it 
is poffible to infcribe in the circle q r s a polygon, 
having an even number of fdes, and greater than the 
ellipfe afbg. Let it be underftood to be infcribed, 
and let a polygon fimilar to it be fuppofed to be in¬ 
scribed in the circle a d b e ; and from the angular 
points of the polygon in a d b e let ftraight lines be 
drawn parallel tODE. Let the points in which thefe 
parallel lines cut the curve of the ellipfe be joined, and 
then a polygon will be infcribed in the ellipfe corre- 
fponding to the polygon in the circle a d b e, as in 
the laft Propofition ; and ab : fg : : the polygon in¬ 
fcribed in the circle a d b e : the polygon infcribed in 
the ellipfe. But, by hypothefis and inverfon, the cir¬ 
cle A D b e : the circle q r s : : a b : f g ; and there¬ 
fore (n. v.) the polygon infcribed in thecircle adbe : 
the polygon infcribed in the ellipfe : : the circle 
adbe: the circle a n s. Confequently (i. and 2. 
xii.) the polygon infcribed in the circle adbe: the 
polygon infcribed in the ellipfe : : the polygon in¬ 
fcribed in the circle adbe: the polygon infcribed in 
the circle a r s. The polygon (14. v.) infcribed in 
the ellipfe is therefore equal to the polygon infcribed 
in the circle a r s : which is abfurd ; for, by the pre- 
fent hypothehs, the polygon infcribed in the circle 
a rs is greater than the ellipfe. 

Secondly, if it be poffible, let the circle a n s be lefs 
than the ellipfe. Then it is poffible, by Cor. Prop. 
XXII. to infcribe in the ellipfe a polygon greater than 
the circle a r s, and a polygon corresponding to it in 
the circle adbe; and to infcribe in the circle qrs 
a polygon f milar to the polygon infcribed in the circle 
adbe. Let fuch polygons be fuppofed to be fo in¬ 
fcribed. 



OF THE ELLIPSE AND HYPERBOLA. 

fcribed. Then it may be dcmonftrated, as above, that BOOK 
the polygon infcribed in the ellipfe is equal to the po- 
lygon infcribed in the circle a r s, contrary to the con- 
llrudion which has now been fuppofed to be made. 

The circle aus therefore is equal to the ellipfe afbg, 
and therefore the ellipfe a f b g is to the circle adbs 
as f g to a b. 

Cor. i. An ellipfe is equal to a circle, whofe diame¬ 
ter is a mean proportional between its axes. For, by 
the above, f g : A b : : the ellipfe A f b g, or the cir¬ 
cle a r s : the circle adbe. But (i. vi.) f g : A b :; 
f G X a e : a b 2 ; and (2. xii.) the circle qrs: the 
circle a d b e : : q s 2 : A b 2 , a s being the diameter 
of the circle a r s. Confequently (n. v.) the ellipfe 
afbg: circle a d b e : : a s 2 : a b 2 : ; f g X a b : 
a e 2 : and (14. v.) a s 2 is equal to f g X a b. 

Cor. 2. From the preceding (and 17. vi. and 2. xii.) 
it is evident, that the areas of two ellipfes are to one 
another as the redangles under their axes. 

The Cor. to Prop. XIV. Lib. I. of the Principia de¬ 
pends, in a great degree, upon this truth. 

PROP. XXIV. 

If from a point in the conjugate axis of an ellipfe a Jlraight 
line, equal to the difference of the femiaxes, he drawn 
to a point in the tranfverfe axis, and he produced be¬ 
yond the tranfverfe axis, fo that the part produced be 
equal to the femiconjugate axis, the extremity of the 
part produced will be in the curve of the fe&ion. Or, 
if from a point in the conjugate axis of an ellipfe a 
jlraight line, equal to the Jum of the femiaxes, be drawn 
to a point in the tranfverfe axis, and if this line be fo 
cut that the fegment between the tranfverfe axis and the 

point 





95 Of the ellipse and hyperbola. 

BOOK. point of feBlon be equal to the femico?ijugate axis , the 
point of feftion will be in the curve . 

Fig. 77. Let a d b e be an ellipfe, of which a b is the tranf- 
7S * verfe, and d e is the conjugate axis, and c is the cen¬ 
ter. Let f be a point in the conjugate, and g be a 
point in the tranfverfe axis, and let the flraight line 
f g be equal to the difference or fum of c b, c d. 
When f g is equal to the difference of c b, c d, as in 
Fig. 78. let f g be produced beyond a b to h, fo that 
g h be equal to c d; but when f g is equal to the 
fum of c b, c d, as in Fig. 77. let the fegment g h be 
equal to c d ; and in either cafe the point h will be 
in the curve of the ellipfe. 

For through the center c draw the flraight line c K 
parallel to fh. Through h draw the flraight line 
h k parallel to d e, and let it meet c k in k, and A B 
in 1. Then (34. i.) the flraight line c k is equal to 
f h. But as f g is equal to the fum or difference of 
c b, c d, and as g h is equal to c d, the flraight line 
F h is equal toe b; and confequently ck is equal to 
c b. With c, therefore, as a center, and c b as a dif- 
tance, let a circle be deferibed, and it will pals through 
k. Again, on account of the fimilar triangles c k 1, 
G H 1, c k 2 : g h 2 : : k i 2 : h i 2 ; and therefore, on 
account of the equals, c b 2 : c d 2 : : k i 2 : h i 2 . 
But (3. and 35. iii.) the fquare of k 1 is equal to the 
re£langle under a 1, 1 b ; and therefore c b 2 : c d 2 : : 
a 1 X 1 b : h i 2 . The point 11 is therefore in the 
curve, by Cor. 1. Prop. V. (and 9. v.) for h i is paral- 
let to the ordinates of a b. 

SCHOLIUM. 

The infirument called by fome the trammels , and by 

others 



OP THE ELLIPSE AND HYPERBOLA. 

others the elliptic compaJJes y ufed by cabinet-makers^ 
&c. for defcribing the curves of ellipfes, are conftru&ed 
on the property demonftrated in this Propofition. As 
the trammels are in general ufe, it is needlefs to give a 
defcription of them in this place. Lathes for making 
pi&ure-frames, and ornaments of an elliptical form, 
are conflrufted on the fame property. 


93 

BOOK 

II. 


A GEO- 





























: i 


























t 




































■ 

















































































































































■■■ - 


















A 


GEOMETRICAL TREATISE 


OF 

CONIC SECTIONS. 


BOOK III. 

Of the Parabola , the Dire Brices cf the SeBions , the Afy imp¬ 
utes of the Hyperbola , Conjugate Hyperbolas , zz-W 0/ 
hyperbolic SeBors and Trapezia. 


DEFINITIONS. 

I. 

HP 

JL HE fe< 51 ion fdc being a parabola, and vbe its - Fig. 7$. 
vertical plane, as in the 15th and 16th Definitions in 
the firft Book, any ftraight line, as d i, in the parabola 
parallel to v b, the fide in which vbe touches the 
cone, is called a Diameter of the parabola. 

Cor. 1. From this Definition (and 9. xi.) the diame- 
. ters of a parabola are parallel to one another ; and, by 
Prop. XIV. Book I. any ftraight line drawn in the 
plane of a parabola, parallel to a diameter, will meet 
the curve in one point, and in one point only, and, by 
this Definition, it will itfelf be a diameter. 


Cor. 









OF THE PARABOLA. 


K Cor. 2 . Any ftraight line in a parabola, not parallel 
to a diameter, will meet the curve in two points. For 
any ftraight line drawn through v, the vertex of the 
cone, and in the vertical plane, and not in the fame di- 
re£lion with v b, will fall without the oppofite cones ; 
and by the demonflration of the fecond part of Prop. 
VIII. of the firft Book, one plane may be drawn 
through this ftraight line to touch the conical fuper- 
ficies, and cut the plane of the parabola. The inter- 
fe&ion alfo of this plane with the plane of the para¬ 
bola will touch the parabola, and any ftraight line in 
the parabola parallel to this tangent will meet the 
curve in two points, by Cor. j. Prop. VIII. Book I. 
Hence (i 6 . xi.) the Cor. is evident. 

IT. 

The point in which a diameter of a parabola meets 
the curve is called the Vertex of the diameter. 

III. 

If a ftraight line terminated by the curve of a para¬ 
bola be bife&ed by a diameter, it is called a Double Or¬ 
dinate to that diameter j and its half is fimply called an 
Ordinate to it. 

IV. 

The fegment of a diameter between its vertex and 
an ordinate is called an slbfcifs of that diameter. 

V. " 

The diameter of a parabola, which cuts its ordinates 
at right angles, is called the Axis of the parabola. 

VI. 

A third proportional to an abfcifs of a diameter of a 
parabola, and the correfponding ordinate, is called the 
Parameter, or Latus Rettum of the diameter. The pa¬ 
rameter of the axis is frequently called the Principal 
Parameter , or Latus Re&um , 


PROP. 




OF THE PARABOLA. 


97 

PROP. I. book 

If each of two diameters of a parabola meet a Jlraight - 

line , and if each of thefe Jlraight lines cut, or one of them 
cut and the other touch the parabola , and if thefe two 
Jlraight lines be parallel 5 then the J'egment of the one 
diameter, between its vertex and the line which it meets, 
will be to the fegment of the other between its vertex 
and the line which it meets, as the fquare of the line 
which meets the firjl mentioned diameter if a tangent, 
or the rectangle under its fegments if a fecant, to the 
fquare of the line which meets the other diameter if a 
tangent, or the reft angle under its fegments if a fecant . 

Suppofe d 1, g k to be two diameters of a parabola, Fig. 80, 
and let d be the vertex of the one, and g the vertex of 
the other. Let l p, m n be two parallel ftraight lines, 
and let d i meet l p in l, and g k meet m n in m, and 
let L p, m n either both cut, or one of them cut and 
the other touch the parabola; then D l is to gm as 
the fquare of l p, if a tangent, or the rectangle under 
its fegments if a fecant, to the fquare of m n, if a tan¬ 
gent, or the re£blngle under its fegments if a fecant. 

For let f g d c be the parabola as formed in the Fig. 79. 
cone, and D 1, g k the diameters mentioned above. 

Let the parabola cut the plane of the bafe in the ftraight 
line f-k 1 c, and let the vertical plane cut it in b e, y B 
being the fide along which the vertical plane touches 
the cone. Then d 1, g k are parallel to V b, by the 

firft Definition. Through the parallels v b, d 1 let a 

plane pafs, and let it cut the cone in the fide v d a, 
and the bafe in b i a. Through the parallels v b, g k 
let a plane pafs, and let it cut the cone in the fide 
v g h, and the bafe in b k ii. Then (4. vi. and 16. v.) 

D 1 : v B : : A 1 : a b, and 

v B : G K : : H B : II it. 

H 


Hence 



OF THE PARABOLA, 


Hence (i. vi.) di : V b •* : ai xib : abxir 
and vb:gk::hbXkb:hkxke. 
But (35. iii.) A 1 x 1 b is equal to f 1 X 1 c, and H K 
x k b is equal to f k x k c ; and, by the feventh 
Lemma, a b x i b is equal to h b X k b. Confe- 
quently, by the above and fubftitution, we have the 
two following ranks of magnitudes proportionals, taken 
two and two in the fame order, 

D 1 : V B : G K 

fixic:abxib:fkxkc; and therefore 
(22. v.) D 1 : G k : ; f 1 X 1 c : f K X k c. 

Let the flraight line fkic have the fame fituation in 
the parabola in Fig. 80. as in Fig. 79. and firft fuppofe 
l p, m n to be parallel to the bafe of the cone, or to 
f c. Then, by Prop. XVI. Book I. d l is to d i as 
the fquare of l p, if a tangent, or the rectangle under 
its fegments, if a fecant, to f 1 x ic; and, by the 
above, di is to g k as f i x i c to f k x k c ; and 
again, by Prop. XVI. Book I. g k is to g m as f k x 
k c to the fquare of m n if a tangent, or the rectangle 
under its fegments if a fecant. We have therefore 


D L 

: d 1 : g k 

g m 

t. L P 2 -| 


rt. m 

or > 

:fixic:fkxkc:- 

<[ or 

f. L P r J 


If. M N r 


Confequently (22, v.) d l is to g m as the fquare of 
l p if a tangent, or the redangle under its fegments if 
a fecant, to the fquare of m n if a tangent, or the red¬ 
angle under its fegments if a fecant. Laftly, let l p, 
m n not be parallel to the bafe of the cone, but let l s, 
m r be parallel to the bafe, and let them touch or cut 
either of the conical luperficies. Then, by the above, 
d l is to g m as the fquare of l s if a tangent, or the 
redangle under its fegments if a fecant, to the fquare 
of m r if a tangent, or the redangle under its fegments 

if 



OB' THE PARABOLA. 


if a fecant. But, by Prop. XIT. Book I. the fquare of BOOK 
l s if a tangent, or the re&angle under its fegments if 111 * 
a fecant, is to the fquare of m r if a tangent, or the 
re&angle under its fegments if a fecant, as the fquare 
of l p if a tangent, or the re&angle under its fegments 
if a fecant, to the fquare of m n if a tangent, or the 
re&angle under its fegments if a fecant. Confequently 
(n. v.) d l is to g M as the fquare of l p if a tangent, 
or the re&angle under its fegments if a fecant, to the 
fquare of m n if a tangent, or the rectangle under its 
fegments if a fecant. 

prop. ir. 

A diameter of a conic feElion bifeEts any firaight line it 
meets in the feElion parallel to a tangent pafjing through 
its vertex ; and ordinates to a diameter , and a tangent 
faffing through its vertex , are parallel to one another . 

In the ellipfe and hyperbola this has been proved, 
according to Cor. 3. to Prop. III. Book II. In the 
parabola aec let the diameter b g cut the ftraight Fig. 
line A c in the point g, and let a c be parallel to de 
touching the parabola in b, the vertex of the diameter 
B G ; the ftraight line A c is bife£led in g. On the 
contrary, any ftraight line in the parabola bife£ted by 
b g is parallel to a c, or the tangent d e. 

Part I. Through a, c let a d, c e be drawn parallel 
to b g, and let a d meet the tangent in d, and c e 
meet it in e. Then, by Cor. j . Def. I. a d, c e are 
diameters, and, by Prop. I. ad : ce : : db 2 : be 2 ; 
and (34. i.) as ad,ce are equal, it follows that d b, 
b e are equal to one another. Confequently (34. i.) 
a g is equal to g c. 

Part II. If it be poffible, let the ftraight line H r in 
the parabola abc be bife&ed by the diameter bg, and 
not be parallel to a c or d e. 

h % 


Through 



IOO 


OF THE PARABOLA. 


book Through r draw r m parallel to the diameter b g* 
!I1 * and through h draw n l parallel to a c or d f., and let 
it meet b g in k, the curve again in l, and r m in m. 
Let b G meet h r in n. Then as h r is bife&ed in x, 
and as k n, m r are parallel, (2. vi.) h n : n r : : h k : 
K M, and h k is equal to k m. But, by Part I. h k is 
equal to K l, and therefore k l is equal to km; which 
is abfurd. Confequently no ftraight line in the para¬ 
bola, unlefs it be parallel to d e, or to the ordinate 
a c, can be bife£ted by the diameter b g. Ordinates 
to the diameter b g mu ft therefore be parallel to one 
another, and to the tangent d e, palling through the 
vertex. 

Cor. 1. From hence, and Prop. III. Book II. it is 
evident, that if a ftraight line be an ordinate to a dia¬ 
meter, any ftraight line in the fe&ion, or oppolite fec- 
tion, and parallel to it, will be an ordinate to the fame 
diameter. 

Cor. 2. From this Propolition it is evident, that if a 
ftraight line bife£t two parallel lines in a conic fe&ion, 
it will be a diameter. 

Cor. 3. From the above a method of finding a dia¬ 
meter of a given parabola is evident. For two parallel 
ftraight lines being drawn in the parabola, a ftraight 
line bife&ing them, and any ftraight line parallel to it, 
will be a diameter. 

Cor. 4. The method of finding the axis of a para¬ 
bola is alfo evident from the above. For, having found 
a diameter of the parabola, by the preceding Cor. let 
a ftraight line at right angles to it be drawn within 
the parabola, and limited both ways by the curve. 
Then the diameter bifecting this ftraight line will be 
the axis; for, being parallel to the diameter firft found, 

- it will (29. i.) bife& the ftraight line in the fe&ion at 
right angles. 


PROP. 



PlateXIl page 100. 



J.Basire sc. 



























































s 








OF THE PARABOLA. 


PROP. III. 

The abfeiffes of a diameter of a parabola are to one another 
as the fquares of the correfponding ordinates; and the 
fquare of an ordinate to a diameter of a parabola is equal 
to the rediangle under the parameter of the diameter , 
and the abfcifs correfponding to the ordinate . 

Part I. Let b g be a diameter of the parabola abc, 
and let a g, h k be ordinates to it, and let them meet 
it in the points g, k, and let b be the vertex of the di¬ 
ameter ; the abfcifs b g is to the abfcifs bk as the 
fquare of the ordinate a g to the fquare of the ordinate 

II K. 

For, as b g is parallel to a fide of the cone, in which 
the fe&ion was formed, and as a g, h k, by Prop. II. 
are parallel, and as they would be bifeded in g, k if 
limited by the curve, this part is evident from Prop. 
XVI. Book I. * 

Part IT. The reft remaining as above, let the ftraight 
line p be a third proportional to the abfcifs b g and 
the correfponding ordinate A g, and confequently the 
parameter to the diameter e g, according to the fixth 
Definition ; the fquare of the ordinate h k is equal to 
the redangle under p and the abfcifs e k. 

For, by the preceding part, e g : b k : : A g 2 : H K 2 , 
and therefore (i. vi.) p x b g : p X b k : : a g 2 : h k 2 . 
But (17. vi.) p x b g is equal to a g 2 , and confequent¬ 
ly (14. v.) p x b k is equal to h k 2 . 

Cor. 1. If a ftraight line touching a parabola meet a 
diameter, the fquare of the fegment, between the point 
of contact and the point of concourfe, will be equal to 

* From this and the fecond part of Prop. II. writers or. projectiles 
prove, that, if the refiftance of the air have no perceptible effect, a pro¬ 
jectile jnuft move in the curve of a parabola. 

H 3 th« 


161 

BOOK 

III. 


Fig.. 



10 z 


OP THE PARABOLA. 


BOOK 

III. 


the rectangle under the fegment of the diameter, be¬ 
tween its vertex and the point of concourfe, and the 
parameter of the diameter, to whofe ordinates the tan¬ 
gent is parallel. For let d b, touching the parabola in 
b, meet the diameter a d in d, and let a g parallel to 
d b meet the diameter b g in g, and let p be the para¬ 
meter of b g. Then (34. i.) a g is equal to d b, and 
A d is equal to b g, and, by Prop. II. d b is parallel to 
the ordinates of B G ; and as, by the above, b g X P is 
equal to A g 2 , b d 2 is equal to A D X p, 

Cor. 2. If a ftraight line cutting a parabola meet a 
diameter, the rectangle under its fegments, between 
the point of concourfe and the curve, will be equal to 
the re&angle under the fegment of the diameter be¬ 
tween its vertex and the point of concourfe, and the 
parameter of the diameter, to whofe ordinates the fe- 
cant is parallel. For, the reft remaining as in the pre¬ 
ceding Cor. let the tangent b d be parallel to the 
ftraight line H m, cutting the parabola in h, l, and 
meeting the diameter r m in m. Then, by Prop. I. 
ad :rm: : db 2 : hm X m lj and therefore (1. vi.) 
A D X p : r,m X p : : d b 2 : h m X M l. Confequent- 
ly, by the preceding Cor. (and 14. v.) r m X p is equal 
to H M X M L. 

SCHOLIUM. 

On account of the equality of the fquare of h k to 
the re&angle under p and b k, Apollonius called the 
fe£lion a parabola. 

From the property demonftrated above the parabola 
is frequently denoted by an algebraical equation, in the 
following manner. Put the parameter of the diameter 
b k = p, the abfeifs b k = x, and the ordinate h k =y. 
Then, from the above, x :y : '-y : p, and p x — y z . 


PROP. 



OP THE PARABOLA. 


I03 


PROP. IV. 

If each of two Jlraight lines meeting one another touch or 
cut , or one of them touch and the other cut , a parabola , 
the fquare of the frjl of the tzuo If a tangent , or the 
rectangle under its Jegments if a fecant, will he to 'the 
fquare of the fecond if a tangent , or the rectangle un¬ 
der its fegments if 'a J'ecant , as the parameter of the di¬ 
ameter to whofe ordinates the jirfl is parallel , to the pa¬ 
rameter of the diameter to whofe ordinates the fecond is 
parallel , 

For, firft let the ftraight lines a e, c E, meeting one 
another in e, touch the parabola in a and c, and let 
e b be a diameter palling through e. Then, by Cor. 1. 
Prop. III. the fquare of a e is equal to the redangle 
under b e, and the parameter of the diameter to whofe 
ordinates a e is parallel; and the fquare of c e is equal 
to the redangle under b e, and the parameter of the 
diameter to whofe ordinates c e is parallel. Confe- 
quently (1. vi.) the fquare of a e is to the fquare of 
c e, as the parameter of the diameter to whofe ordi¬ 
nates a e is parallel to the parameter of the diameter 
to whofe Ordinates c e is parallel. 

Next, let the ftraight line g a, touching the parabola 
in a, meet in g the ftraight line g k, which cuts the 
parabola in f, k, and let g p be a diameter palling 
through g. Then, by Cor. 1. Prop. III. the fquare of 
A g is equal to the redangle under d g, and the para¬ 
meter of the diameter to whofe ordinates a g is paral¬ 
lel; and, by Cor. 2. Prop. III. the redangle kgf is 
equal to the redangle under D g, and the parameter of 
the diameter to whole ordinates gk is parallel. Con- 
fequently (1. vi.) the fquare of a g is to the redangle 
K g f as the parameter of the diameter to whofe ordi- 
h 4 nates 


BOOK 

111. 


Fig. 83. 



OF THE PARABOLA. 


I04. 

BOOK 

III. 


Fig. 83. 


nates a g is parallel, to the parameter of the diameter 
to whofe ordinates G K is parallel. 

Laiily, if the ftraight line g k, cutting the parabola 
in f, k, meet in the point g the ftraright line g l, 
which cuts the parabola in the points h, l, then it may 
be proved in the fame way, by means of Cor. 2. Prop. 
III. that the rectangle k g f is to the rectangle l g h 
as the parameter of the diameter to whofe ordinates g k 
is parallel, to the parameter of the diameter to whofe 
ordinates g l is parallel. 

PROP. V. 

If a firelight line touching a parabola meet a diameter , and 
an ordinate to the diameter pafs through the point of con- 
tail) the fegment of the diaineter , between its vertex and 
the tangent , will be equal to its abfeifs , between its ver¬ 
tex and the ordinate . 

Let the ftraight line A e, touching the parabola A b c 
in the point a, meet the diameter b d in the point e, 
and through the point of contact A let the ordinate A d 
to b d pafs, and meet e d in d 3 the fegment b e be¬ 
tween b the vertex and the tangent is equal to the ab¬ 
feifs b d between the vertex and the ordinate. 

For produce a d till it meet the curve in c, and draw 
c f parallel to e d, and let it meet a e in f. Then, 
by the third Definition, a c is bife&ed in d, and, by 
Cor. 1. to the firft Definition, c f is a diameter 3 and, 
by Prop. I. b e : c f : : a e 2 : a F 2 . But c f, d e be¬ 
ing parallel, (2. vi.) ad:dc::Ae:ef, and A c be¬ 
ing bife&ed in d, a f is bife&ed in e, and for the fame 
reafons d e is half of c f. Confequently A f 2 is equal 
to four times a e 2 (4. ii.) and, therefore, c f is equal 
to four times b e 3 and as c f is double of d e, b e is 
equal to b d. 

Cor. 



OP THE PARABOLA. 


Cor. If A g, b d be any two diameters of the para¬ 
bola arc, and if a d be an ordinate to R D, and b g 
be an ordinate to ag, the abfciflfes a g, b d will be 
equal. For let a e touch the parabola in a, and meet 
the diameter b d in e. Then, by Cor. i. to the firft 
Definition, a g, e b are parallel, and, by Prop. II. ae, 
g b are parallel. Confequently (34. i.) a g is equal 
to e b, and therefore, as by the above eb, bd are 
equal, ag is equal toBD. 

PROP. VI. 

If two flraight lines touching a conic fiction , or oppcfte 
hyperbolas, meet one another, the diameter hifeding the 
line joining the points of contact will paj's through the 
point of concourfe. 

I11 the ellipfe, hyperbola, or oppofite hyperbolas, this 
has been proved in Prop. VIII. Book II. In the para¬ 
bola ab.c let the two ftraight lines e A, e c touch the 
fection in the points a c, and meet one another in e, 
and let the diameter b d bifedt a c, the ftraight line 
joining the points of contact in d ; the diameter b d 
will pafs through e. 

For, as a c is bifedted by the diameter b d, it is a 
double ordinate to bd; and therefore, by Prop. V. 
if b d be produced and meet the tangents, its fegment 
between b the vertex and the tangent a e will be 
equal to its abfeifs b d ; and its fegment between b 
and the tangent c e will alfo be equal to b d. The 
diameter b d will therefore meet both the tangents 
a e, c e in the fame point, and confequently will pafs 
through e, the point of concourfe. 

Cor. 1. If two ftraight lines touching a conic fedlion, 
or oppofite hyperbolas, meet one another, a ftraight 
line palTing through the point of concourfe, and bifecl- 

ing 


I0 5 

BOOK 

III. 


Tig. Sj. 



io6- 


OF THE PARABOLA. 


BOOK 

III. 


FiS. 84. 


ing the line joining the points of contaft, will be a di¬ 
ameter. The truth of this is evident from Cor. Prop. 

VIII. Book II. and the above. 

Cor. 2. From the above it is evident, that if a c be a 
double ordinate toB d, a diameter of any conic lec¬ 
tion a bc, and if A e, touching the fedtion in a , meet 
the diameter in e, then if the ftraight line ec be 
drawn, it will touch the fe£tion in c. 

DEFINITIONS. 

VII. 

If from b, the vertex of the axis a b of the parabola 
v B M, a fegment b f be taken in the axis equal to one 
fourth of the parameter of the axis, the point f is 
called the Focus, or Umbilicus , of the parabola. 

Cor. The double ordinate t s, drawn through f the 
focus of any conic fe6tion, is equal to the parameter of 
the axis paffing through the focus. This has been 
proved in the ellipfe and hyperbola in Cor. 4. to the 
eleventh Definition in Book II. In the parabola the 
fquare of t f is equal to the rectangle under b f, and 
four times b f, by thisDef. and Prop. III. and therefore 
4 T F 2 is equal to 4 b f x 4 e f. But (4. ii.) 4 t f 3 is 
equal to t s 2 , and confequently t s is equal to 4 b f, 
and therefore, by this Del. equal to. the parameter of 
the axis a b. 

VIII. 

As in the ellipfe and hyperbola, fo in the parabola, 
the ftraight line touching the fe&ion in t, the extre¬ 
mity of the double ordinate drawn as above, is called 
the Focal Tangent to the parabola. 

IX. 

The tranfverfe axis of an ellipfe or hyperbola, and 
the axis of a parabola, is fomctimes called the Focal 
4 xis of the lection. 

X. If 



OF THE DIRECTRICES OF THE SECTIONS. 


IO7 

X. BOOK 

If the focal tangent t g, belonging to the focus f in 1IL 
any conic fe&ion p b m, meet the focal axis a b in x, 
the ftraight line x y at right angles to a b is called a 8 5- 
Directrix of the feftion. And, if in the ellipfe or hy¬ 
perbola o be the other focus, and the focal tangent be¬ 
longing to o meet the focal axis a b in k, the ftraight 
line k l at right angles to a b is alfo called a directrix 
of the ellipfe or hyperbola. 

Cor. 1. As in the ellipfe and hyperbola the foci F, o 
are equally diftant from c the center, it is evident 
from the above, and Prop. VII. Book II. that the di¬ 
rectrices x y, k l are equally diftant from the center. 

Cor. 2. In the parabola, the focus f and the direc¬ 
trix x y are equally diftant from b, the vertex of the 
axis, by the above and Prop. V. 

PROP. VII. 

If a tangent pafjing through the vertex of the focal axis 
cf a conic feClion meet a focal tangent , its Jegment be¬ 
tween the 'point of contad and point of concourfe will 
he equal to the Jegment of the axis between the point of 
contact and the focus to which the focal tangent be¬ 
longs. 

Let the tangent b g, palling through b the vertex of Fig. 
the focal axis a b, of any conic fe£tion pbm, meet in gf* 
the point g the focal tangent t g belonging to the fo¬ 
cus F ; the fegment g b is equal to the fegment f b. 

And in the ellipfe and hyperbola the tangent A h, 
palling through a the other vertex of the focal axis, 
and meeting the focal tangent t g in h, is equal to the 
fegment a f. 

As far as this Propolition relates to the ellipfe, or hy¬ 
perbola, it has been proved in Prop. XII. Book 11 . 

In 


ioS 


BOOK 

III. 


Fig. 84. 

8«f. 

86 . 


OF THE DIRECTRICES OP THE SECTION'S. 

In the parabola, every thing remaining as in the 
feventh Definition and its Cor. t f is double of f b, 
and therefore, by Cor. 2. to the tenth Definition, t f 
is equal to f x. Again, by Prop. II. tf,gb are pa¬ 
rallel, and therefore (4. vi.) tf:fx;:gb:bx, and 
g B is equal to b x, and confequently equal to f b. 

PROP. VIII. 

A ft might line drawn from any point in the curve of a co¬ 
nic. fettion to a focus is to a fraight line drawn from 
the fame point perpendicular to the direftrix nearef this 
focus , as the fegment of the axis between the fame focus 
and the nearef vertex, to the fegment between this ver¬ 
tex and the directrix: and, in the ellipfe and hyperbola, 
a fraight line drawn from the fame point in the curve 
to the other focus is to a fraight line drawn from the 
fame point perpendicular to the other directrix in the 
fame ratio . 

A ftraight line p f, drawn from, any point p in the 
curve of the conic fe&ion p e m to the focus f, is to 
p y perpendicular to x y, the directrix neareft to f, as 
the legment f b, of the focal axis between f and the 
vertex b, to the fegment b x of the fame axis between 
x the vertex and the dire&rix: and, in the ellipfe and 
hyperbola, p o drawn to the other focus o is to p a 
drawn perpendicular to the other directrix k l in the 
fame ratio. 

For, the reft remaining as in the preceding Prop, and 
Def. X. through p draw p m an ordinate to the axis a b, 
and let it meet the curve again in m, the axis A b in R, 
and the focal tangent t g in n. Then, by Prop. XIII. 
Book I. t G 2 : t N 2 : : g b 2 : p n X N M. But, by 
Prop. II. n m, t f, G b are parallel, and therefore (10. 
and 22. vi.) t g 2 : t n 2 : : f b 2 : f r 2 ; and therefore 

(ii. 



OF THE DIRECTRICES OF THE SECTIONS* IC9 

(1 1, v.) F b 2 : f r 2 : : G IT : r n X n m. Confequent- Book 

ly* as by Prop. VII. f b is equal to g b, f b 2 is equal_ ^ _ 

to G b 2 , aucl (r4. v.) f r 2 is equal toPNXNM; and 
therefore (6. ii. and 47. i.) r n 2 is equal to p f 2 , and 
R n is equal to p f. But (4. vi.) r n : r x : : g b : 

B x, and therefore as (34. i.) p y is equal to r x, and, 
by Prop. VII. f b is equal to gb,pf:py::fb: 
b x. 

Again, in the ellipfe and hyperbola, the reft remain¬ 
ing as above, let c be the center, and let n v perpen¬ 
dicular to the directrix k l meet k l in v. Then (34. i.) 
nv, p ft are equal, and v k is equal to n r, and con- 
lequently equal to p f. Let c d the femiconjugate 
axis be produced till it meet the focal tangent T g in 
I, and, by Cor. 2. Prop. XIII. Book II. c 1 will be 
equal to c b or c a. Alfo (4. vi.) x c : c 1 : : x k : 

K l ; and therefore, as x c, c k are equal, k l is equal 
to a b the tranfverfe axis. Confequently, by Cor. 1 . 

Prop. XIII. Book II. l v is equal topo; and as l v 
is parallel to g b, and v n to b x, l v : v n : : g b : 

B x. On account of the equals therefore, p o : p q ; : 

F b : b x. 

Cor. 1. In the ellipfe and hyperbola, (4. vi.) c 1 : 
c x : : n r : R x; and therefore on account of the 
equals c b r cx : : p f : p y. But, by Prop. VII. 

Book II. c B : c.x : : c f : c b ; and therefore (1 £. v.) 
cf:cb::pf:py. For the fame rcafons, or by the 
above, (and 11. v.) c f : c b : : f b : b x. 

Cor . 2. By the preceding Cor. in the ellipfe p, f is 
lefs than p y, but in the hyperbola p f is greater than 
p y. And, as by the above (and ii.v.) pf:py:: 

P o : p q, in the ellipfe p o is lefs than p a, but in the 
hyperbola po is greater than p a. 

Cor. 3. A ftraight line drawn from any point in the ^ 
curve of a parabola to the focus is equal to a ftraight 

line 



no 


OF THE PARABOLA. 




book line drawn from the fame point perpendicular to the 
JI1, directrix. For pf;py::fb:bx ; and f b is equal 
to B x. 

SCHOLIUM. 

Some writers on conic fe£tions have chofen this pro¬ 
perty as the primary one for their treatifes, and accord¬ 
ing to it have defined the fe£tions in the following 
manner/ 

Fig. 84. Let f be a point without the ftraight line x y, and 
86. whilft a ftraight line f p revolves about f as a center, let 
a point p fo move in f p that f p may always be to p y, 
perpendicular to x y, in a given ratio. The curve de- 
fcribed by the point p will be a conic fe£Uon ; and it 
will be a parabola, ellipfe, or hyperbola, according as 
F p is equal to, lefs, or greater than p y. 

PROP. IX. 

if from any point in the curve of a. parabola a Jlraight 
line he drawn to the focus , and a Jlraight line perpendi- 
cular to the directrix , the angle contained hy thefeJlraight 
lines will be biJ'cEled by a tangent puffing through the 
fame point. 

Fig. 89. From the point p in the curve of the parabola p b r 
let the ftraight line p f be drawn to f the focus, and 
the ftraight line p d perpendicular to d x the directrix; 
the'ftraight line p e, touching the parabola in p, bife&s 
the angle f p d. 

For let the tangent p e meet the axis a b in e, and 
let p a be an ordinate to a b. Then, by Prop. V. a b 
is equal to b e ; and as, by Cor. 2. to the tenth Defi¬ 
nition, f b is equal to b x, a x is therefore equal to 
F e. But (34. i.) a x is equal to pd ; and, by Cor. 2. 
Prop. VIII. r d is equal to pf. Confequently p f, 

f E 



OF THK l’ARABOLA. 


f i: arc equal, and therefore (5, i.) the angle f r n is 
equal to the angle f k i\ But as P d, a e arc parallel, 
the angle f e p (29. i.) is equal to the angle d p e. 
(’onfequently the angle f p e is equal to the angle 
d p e, arid therefore the angle f p d is bifeded by the 
tangent p e. 

Cor, If a flraight line touching a parabola meet the 
axis, the fegment of the axis between the point of con- 
courle and the focus is equal to the flraight line drawn 
from the point of contact to the focus. This is evident 
from the above, for f e is equal to p f. 

SCHOLIUM. 

It is not certain when, or by whom, the name Focus , 
or Umbilicus , was firft given to a point in a conic fee- 
tion. Neither of the two occurs in the Treatife of Apol¬ 
lonius, or in the writings of Archimedes, who oeca- 
fionally mentions properties of the fe£tions. The points 
themfelves, however, in the ellipfe and hyperbola, 
were well known to Apollonius. He calls them pu?ift<i 
ex applicatione fafla ; and he demonftrates the molt 
important properties of lines related to them. He 
does not mention the focus of the parabola. 

It is highly probable, from analogy of general prin¬ 
ciples, and from the hiftory of this branch of fcience as 
far as it has been traced, that optical purfuits firft fug- 
gefled each of the two names. For, in opticks, if a ray 
of light fall upon a plane furface, the angle of inci¬ 
dence is equal to the angle of reflexion; and if a ray 
of light fall upon a curve, and a flraight line touch the 
curve in the fame point, the angle contained by the 
incident ray and tangent will be equal to the angle 
contained by the refle&ed ray and tangent, as a tan¬ 
gent is the diredion of a curve in the point of con- 
tad. 


in 

BOOK 

in. 


Thefe 



113 


OF THE PARABOLA. 


BOOK Tbefe truths being premifed, fuppofe a conic fection 
1IL rbp to revolve about ab, the focal axis, and that a 

^ 7777 "” concave fpeculum is formed by the curve by this revo- 
90. lution. Let the ftraight line t p e touch any one of 
91 ’ the fe&ions in the point p. In the eliipfe and hyper¬ 
bola let f, o be the foci, and let f be the focus of the 
parabola. 

Fig. 91. 1. Let k P be a ray of light, whofe direction, in a 

firaight line, pafles through o the focus oppofite to f, 
within the hyperbolic fpeculum rbp. Let it fall upon 
the fpeculum in the point p, and draw p f, and pro¬ 
duce k p to o. Then, as by Prop. XV. Book II. the 
angle fpe is equal to the angle ope, and confequent- 
3 y (15. i.) equal to the angle kpt, the ftraight line 
p f will reprefent the ray of light after reflexion. For, 
if a perpendicular to T e be drawn from p, the angle 
contained by it and k p will be equal to the angle con¬ 
tained by it and f p ; the angle of incidence being 
equal to the angle of reflexion. Hence if any number 
of rays fall on a concave hyperbolic fpeculum, and be 
converging to the focus in the oppofite hyperbola, they 
will be reflected to the focus within the fpeculum. 

Fig. 90. 2. Let a r d b p be a concave elliptic fpeculum. Let 

a ray of light proceeding from the focus o fall upon 
the fpeculum at p, and let pf be drawn. Then will 
p F reprefent the ray after reflexion, for the fame rea- 
fons as above, as the angles opt,fpe are equal, by 
Cors-i. to Prop. XV. Book II. Hence it is evident, 
that if any number of rays proceed from one focus of 
an elliptic concave fpeculum, they will be reflected 
into the other. As rays of heat are fubject to the' 
fame laws with rays of light, as to incidence and re¬ 
flexion, if a fire or any heated body be placed at o, 
within the concave elliptic fpeculum, the whole of the 
heat after reflexion will meet at f. Perhaps attention 

to 



OP THE PARABOLA. 

to this property might be of confiderable ufe in fitting 
up fire-places, reverberating furnaces, &c. 

3. Let k p be a ray of light parallel to a b the axis 
of the parabola, by vvhofe revolution the parabolic fpe- 
culum is generated, and let pf be drawn. Then will 
p f reprefent the ray after reflexion. For, k p being 
produced to d, the angle f p e will be equal to the 
angle d p e, by the Propofition preceding this Scho¬ 
lium, and therefore (15. i.) the angle k p t will be 
equal to the angle fpe, Confequently, as above, p f 
is the ray after reflexion. Hence if any number of 
rays, parallel to the axis, fall upon a concave parabolic 
fpeculum they will all after reflexion meet in the 
focus. 

The very confiderable magnifying powers which re¬ 
flecting telefcopes are capable of, with parabolic fpe- 
cula, are to be attributed to this property. For a ce- 
leftial body being at an immenfe diftance, the rays 
which iflue from it upon the parabolic fpeculum are, 
as to fenfe, parallel to the axis; and, being all reflected 
to the focus, a diftinCt and vivid image of the body is 
produced, provided the compofition of the metal be 
good and the parabolic figure juft. 

PROP. X. 

AJlraigbt line drawn from the focus of a parabola , per¬ 
pendicular to a tangent , is a mean proportional between 
the Jlraigbt line drawn from the point of contact to the 
focus , and the fegment of the axis between the focus and 
the vertex of the axis. 

From f the focus of the parabola pb r let the ftraight 
line f G be drawn perpendicular to the ftraight line p e, 
touching the parabola in p, and draw pf; f g is a 
mean proportional between p f and f b, the fegment 

1 




BOOK 

in. 


Fig. 89. 


Fig. 89. 



OP THE PARABOLA, 


114 

book of the axis a b, between f and b the vertex of the 
ill. . 

___ axis. 

For let the tangent p e meet the axis in e, and draw 
B g, and let p a be an ordinate to the axis. Then, by 
Cor. Prop. IX. pf,fe are equal, and therefore (5. i.) 
the angles F p e, f e p are equal. Confequently in 
the triangles fgp,fge, as the angles at g are right 
angles, pg is (26.1.) equal to ge, and the angles 
p f g, e F G are equal. Confequently, as, by Prop. V. 
a b is equal to b e, p g : G e : : A B : b e, and (2. vi.) 
g b is parallel to the ordinate p A, and therefore g b f 
is a right angle. The triangles p f g, g f b are there¬ 
fore equiangular, and (4. vi.) pfifghfgifb. 

The above Prop, is Lemma XIV. Lib. I. of the Prirw 
eipia. 

Cor . 1. Hence (Cor. 2. 20. vi.) p f 2 : f g 2 : : p f : 

F B. 

Cor. 2. The concourfe of any tangent p E with a 
ftraight line f g, drawn from the focus of the parabola 
perpendicular to the tangent, is in the ftraight line b g, 
which touches the parabola in the vertex of the axis. 
For, by the above, g b is parallel to the ordinate p a, 
and therefore the Cor. is evident by Prop. II. 

Cor. 3. If a ftraight line touch a parabola, and cut a 
ftraight line drawn from the focus to the directrix at 
right angles, it will bifedt it. For, the reft remaining 
as above, let f g produced meet the directrix in d. 
Then as gb, dx are perpendicular to the axis, they 
are parallel, and (2. vi.) fb:bx::fg:gd, and as, 
by Cor. 2. Def. X. f b is equal to b x, f g is equal to 
GD. 

SCHOLIUM. 

If a ftraight line pafs through a point moving in the 
curve of a conic fedtion, and always touch the fedtion, 

and * 



0£ THE PARABOLA. 


**s 


BOOK 

III. 


Fig- 89. 
90. 
9 *- 


and if a ftraight line revolve about a focus of the fee- 
tion as a center, and be always perpendicular to the 
moving tangent, the magnitude of the perpendicular 
will be lefs varied in the hyperbola than in the para¬ 
bola, but it will be more varied in the ellipfe than in 
the parabola. 

For let the point p be fuppofed to move in the curve 
b p of the conic fe&ion pbr, and let the ftraight line 
p e accompany it in its motion, and always touch the 
fe&ion. Let the ftraight line f g revolve about f, a 
focus of the fe&ion, and let it be always perpendicular 
to the tangent p e. 

In the ellipfe and hyperbola let c be the center, c d Fig. 90, 
the femiconjugate axis, and c h the femidiameter pa¬ 
rallel to the tangent p e. Then, by Prop. XIX. Book 
II. fg : fp: : cd : ch, and therefore, by Lemma V. 
f g 2 : f p 1 : : c d 1 : c h 2 . But o being the other fo¬ 
cus, and p o being drawn, by Prop. XVIII. Book II. 
ch 2 =fpxpo, and therefore fg j :fp 2 ::cd 2 : 

fp 2 x cd 2 FPXCD 2 


F p x p o, and f g 2 = 


For the 


F P X P O P O 

fame reafons if p denote another pofition of the moving 

point, F g the perpendicular at that pofition, and F />, 

p o ftraight lines drawn from p to the foci, then f^ 2 = 

FfXCD 2 n . . „ , FPXCD- 

Confequently f g* : F g 2 : * - 9 


p o 

b X CD 2 
p O 


P O 


F P 
’ PO 


F J 

P o 


If in the parabola p denote another pofition of the Fig. 89. 
moving point, the perpendicular at that pofition, 
and p Fa ftraight line drawn to the focus, then, by the 
Propofition preceding this Scholium, (and 17. vi.) 

F g 1 — f p X F B, and fg 2 :F£* 2 :: FPXFB : F p x 

TV Tp 
FB ' FB* 


F e : : f p : F p ; or f g : f g z 


1 % 


Iu 








Ji 6 
BOOK 

m. 


OF THE PARABOLA* 

In the hyperbola and ellipfe, therefore, the fquare of 
the perpendicular f g varies as the value of the frac- 

p p p p 

tion — varies, and in the parabola it varies as — 
p o fb 

varies. 

But if the numerator and denominator of a fra&ion 
be each variable, then if they always increafe or de- 
creafe in the fame proportion, the value of the fraction 

will be always the fame. Thus if the fraction be 
and if while o. varies and becomes q, r varies and be¬ 
comes r, and if it be a : r : : q : r, then — = —, by 

converting the proportion into an equation. From 
hence it is alfo evident, that the more nearly the nu¬ 
merator and denominator increafe or decreafc in the 
fame proportion, the lefs will the value of the fraction 
be varied ; but, on the contrary, the more they differ 
from a proportional increafe or decreafe, the more will 
the value of the fraction be varied. Now in the hy¬ 
perbola the difference between f p, p o is, in every li- 
tuation of p, equal to a b the tranfverfe axis, and there¬ 
fore, in every inftant they vary, they receive an equal 
increafe or diminution ; but however, in the parabola, 
F p may increafe or diminifh, f b remains conftant. In 

the hyperbola therefore the value of the fra£tion — ? 

PO 
F P 

varies lefs than the value of the fra&ion — in the pa- 

FB 

rabola. Again, in the ellipfe the fum of pf,po is 
equal to a b the tranfverfe axis, and therefore if either 
of them increafe, the other will diminifli ; and confe- 
quently in varying they will differ more from a pro¬ 
portional increafe at the fame time, or decreafe at the 
fame time, than f p, f b in the parabola. In the el¬ 
lipfe 



OF THE PARABOLA* 


lipfe therefore the value of the fra&ion varies more m° K 

po 

' . F P 

than the value of the fraction — in the parabola. 

FR r 

In the hyperbola therefore the fquare of f g varies 
lefs, but in the ellipfe it varies more, than it varies in 
the parabola ; and confequently f g varies lefs in the 
hyperbola, but more in the ellipfe, than it varies in the 
parabola. See the Principia, Cor. 6. Prop. XVI. Lib. I. 

PROP. XI. 

If from a point in which a fraight line touches a parabola 
two Jlraight lines be drawn to the axis, one of them an 
ordinate to it, and the other at right angles to the tan¬ 
gent, the fegment of the axis intercepted between them 
will be equal to half the parameter of the axis : and if 
a fraight line touching a parabola meet the axis, ajid a 
fraight line, at right angles to the tangent, be drawn 
from the point of contact to the axis, the fegment of the 
axis intercepted between them will be equal to half the 
parameter of the diameter pajfng through the point of 
contafl. 

Part I. From the point p, in which the ftraight line Fi S- * 9 * 
p e touches the parabola p b r, let the two ftraight 
lines p a, p h be drawn to A b the axis, one of them 
p A an ordinate to it, and the other p h at right angles 
to the tangent p e ; the fegment h a of the axis, in¬ 
tercepted between them, is equal to half the parameter 
of tbe axis. 

For let the tangent p e meet the axis in e, and then 
as p a is an ordinate to the axis, it is at right angles to 
h e ; and (Cor. 8 . vi.) the fquare of p a is equal to the 
rectangle under e a, a h. But, by Prop. V. E B, e a 
are equal, and, by Prop. III. the fquare of p A is equal 
13 to 



OF THE PARABOLA. 


II H 

BOOK 

III. 


to the rectangle under b a and the parameter of the 
axis. Consequently the re&angle under E a, a h is 
equal to the rectangle under b a and the parameter of 
the axis, and therefore (i 6. vi.) e a is to b a as the pa¬ 
rameter of the axis to ah; and as b a is half of k a, 
a h muft be half of the parameter of the axis. 

Part II. Let the ftraight line p e touching the para¬ 
bola p h h in p meet the axis a b in K, and let the 
ftraight line e h, at right angles to P E, meet the axis 
a b in H ; the fegment E H, intercepted between p e, 
p H, is equal to half the parameter of the diameter p k, 
pafling through the point of contact. 

For, the reft remaining as in the preceding part, 
the fquare of p e (Cor. 8. vi.) is equal to the rectangle 
under 11 k, e a ; and, by Cor. I. Prop. III. the fquare 
of p k is equal to the re&angle under E b and the para¬ 
meter of p k. Oonfcqucntly the rectangle under h e, 
r a is equal to the rectangle under e b and the para¬ 
meter of p k, and therefore (16. vi.) e a is to eb as 
the parameter of p k to h b ; and as e b is half of e a, 
ji k muft be equal to the half of the parameter of p k. 

Cor, i. The parameter of the axis is lei’s than the 
parameter of any other diameter. 

Cor. 2. A ftraight line drawn from any point in the 
curve of a parabola to the focus is equal to a fourth 
part of the parameter of the diameter palling through 
the fame point. If the point be the vertex of the axis, 
this is evident from the feventh Definition; but for 
any other point p let every thing remain as in this 
Prop, and draw p p to the focus F. Then, by Cor, 
Prop. IX. F p, f e arc equal; and asEPH is a right 
angle, if with f as a center, and f p as a diftance, a cir¬ 
cle be described, it will pafs (31. Hi.) through e and h, 
Confequcntly k r, f p, fh are equal, and therefore, by 
part II. of this Prop, each of them is equal to a fourth 

part 



OS' THE PARABOLA. 


part of the parameter of p k. This is Lemma XIII. book 
L ib. T. of the Principia. ’ 

Cor. 3. The diftance of the vertex of any diameter 
of a parabola from the directrix is equal to a fourth 
part of the parameter of the diameter. This is evident 
from the preceding Cor. and Cor. 3. Prop. VIII. 

PROP. XII. 

Any Jlraight line drawn through the focus of a parabola , 
and terminated both ways by the curve , is equal to the 
parameter of the diameter to which it is a double ordi- 
7iate . 

Let the ftraight line a h, palling through f the fo- F5 S- 8 7 * 
cus of the parabola abc, meet the curve in a and H, 
and be a double ordinate to the diameter d £; a h is 
equal to the parameter of d e. 

If d e be the axis, the Prop, is the fame as the Cor. 
to the feventh Definition. Let d f. therefore not be 
the axis, and let it meet ah in e. Let d g touch 
the parabola in d the vertex of d e, and meet b f the 
axis in g, and draw d f. Then, by Cor. 1. to the firft 
Definition, and Prop. II. g e is a parallelogram, and 
therefore (34. i.) d e, g f are equal. But, by Cor. 

Prop. IX. g f, d f are equal, and, by Cor. 2. Prop. XI. 
d f is equal to a fourth part of the parameter of de. 
Confequently d e is equal to a fourth part of the para¬ 
meter of d e, and therefore, by Prop. III. a e 2 is equal 
to d e x 4 d e, and 4a e 2 is equal to4D e X 4DE. 

But (4. ii.) 4 a e 2 is equal to a h 2 , and therefore A H 
is equal to 4 d e, and confequently equal to the para¬ 
meter of D E. 

Cor . If a ftraight line in a parabola pafs through the 
focus, and cut the diameter to which it is an ordinate, 
the abfoifs of the diameter will be equal to tfye diftance 

14 of 



120 


OF THE PARABOLA. 


BOOK 

III. 


Fig. 88. 


Fig. 88. 


of its vertex from the focus, and alfo from the direc¬ 
trix. For, as above, d e is equal to d f, and therefore, 
by Cor. 3. Prop. VIII. the remaining part of this is 
evident. 

DEFINITIONS. 

XI. 

The fuperficies aclbia ; bounded by the ftraight line 
A c and the curve A 1 b l c of a parabola, is called an 
interior parabolic fegment; and if the ftraight lines A e, 
c e, touching the parabola in A, c, meet one another 
in E, the fuperficies bounded by ae, c e, and the curve 
A 1 b l c, is called the exterior parabolic fegment , corre- 
fponding to the interior fegment firft mentioned. 

XII. 

If the ftraight line r s, parallel to A c, touch the pa¬ 
rabola in b, and meet in the points r, s the diameters 
a r, cs, the parallelogram arsc is faid to be cir- 
cumfcribed about the interior parabolic fegment 

A I B L C A. 

PROP. XIII. 

In an interior parabolic fegment a rectilineal figure m.ay 
be infcribed , and a correfponding rectilineal figure may 
be infcribed in the exterior fegment, fa that the reCii- 
lineal figure in the interior fegment Jhall be double of 
that infcribed in the exterior ; and thcfe rectilineal in¬ 
fcribed figures may be fuch , that each fioall be lefs than 
the fegment in which it is infcribed by a fuperficies lefs 
than any given fuperficies . 

Let a c l b 1 A be an interior, and eaiblce the 
correfponding exterior parabolic fegment, as in the 
eleventh Definition ; a rectilineal figure may be in¬ 
fcribed in the fegment acle i a, and a correfponding 

reCti- 



OF TI1E PARABOLA. 


121 


re&ilineal figure may be inferibed in the fegment book 
e A i b l c e, fo that the rectilineal figure in the interior ]t1, _ 
fegment (liall be double of that in the exterior; and 
thefe reClilineal figures may be fuch, that each (liall be 
lofs than the fegment in which it is inferibed by a fu- 
perficies lefs than a given fuperficies o. 

For bifect a c in d, and draw e d, and let it cut the 
curve in b. Let the ftraight line f g touch the para¬ 
bola in b, and meet the tangent e a in f, and the tan¬ 
gent e c in g, and draw ab, c b. Then* by Cor. i. 

Prop. VI. e d is a diameter of the parabola, and, by 
Prop. II. f G, ac are parallel; and, by Prop. V. d b, 
b e are equal. Confequently (2. vi.) a f, f e are equal, 
and (1. vi.) the triangle afb is equal to the triangle 
E f e ; and therefore, as (1. vi.) the triangle A bd is 
equal to the triangle a e b, the triangle abd is double 
of the triangle efb. For the fame reafons the trian¬ 
gle c b d is double of the triangle e g b ; and there¬ 
fore the triangle abc, inferibed in the interior para¬ 
bolic fegment, is double of the triangle ef g, inferibed 
in the exterior parabolic fegment. Again, bifeCf a b 
in h, and draw f h, and let it meet the curve in 1. 

Let the ftraight line m n touch the parabola in 1, and 
meet the tangent a f in m, and the tangent f b in n, 
and draw a i, i b. Then, as above, it may be proved, 
that the triangle aib, inferibed in the interior para¬ 
bolic fegment A 1 b a, is double of the triangle f m n, 
inferibed in the exterior parabolic fegment faibf. 

Alfo bifeCl b c in k, and draw g k, and let it meet the 
curve in l. Let the ftraight line p a touch the para¬ 
bola in l, and meet the tangent c G in a, and the tan¬ 
gent b g in p, and draw b l, lc. Then it may be 
alfo proved, as above, that the triangle b l c, inferibed 
in the interior parabolic fegment e l c b, is double of the 



122 


OF THE PARABOLA. 


BOOi 
III. ' 


triangle g p q, infcribed in the exterior parabolic leg¬ 
men t g b l c g. Confequently the re&ilineal figure 
aiblc, infcribed in the interior parabolic fegment 
firft mentioned, is double of the re&ilineal figure 
E m n p a, infcribed in the correfponding exterior pa¬ 
rabolic fegment firft mentioned. In the fame manner 
the infcription in each fegment may be continued, fo 
that the whole figure infcribed in the interior fegment 
fhall be double the whole figure infcribed in the exte¬ 
rior fegment. 

The infcription may alfo be continued till the recti¬ 
lineal figure fhall be lefs than the fegment in which it 
is infcribed by a fuperficies lefs than the fuperficies o. 
For in the interior fegment the triangle abc, being 
equal to half the triangle eac, is greater than half the 
fegment aiblca; and for the fame reafons the trian¬ 
gle a i b is greater than half the fegment a i b a, and fo 
of other remaining fegments. Alfo in the exterior feg¬ 
ment the triangle e f g, being half of the reCtilineal 
figure e a b c e, is greater than half the fegment 
e a i b l c e ; and for the fame reafons the triangle fmn 
is greater than half the fegment faibf, and fo of other 
remaining fegments. Confequently (i. x.) the infcrip¬ 
tion may be continued till each of the re&ilineal fi¬ 
gures (hall be lefs than the fegment in which it is in¬ 
fcribed by a fuperficies lefs than the given fuperfi¬ 
cies o. 

Cor. The tangent f g being produced till it meet 
the diameters a r, c s in n and s, the circumfcribed 
parallelogram R a c s is equal to the triangle eac, 
and therefore equal to the interior and exterior para¬ 
bolic fegments taken together. For, by the above, 

E F, fa are equal, and as r a, e b are parallel, the 
triangles e f b, a f r are equiangular, and (26. and 

4 .i.) ' 



OP THE PARABOLA. 


I23 


4. i.) therefore equal. For the fame reafons the tri- BOOK 
angles e g b, c g s are equal; and consequently the in * 
Cor. is evident. 

PROP. XIV. 

An interior parabolic fegment is double of the correfponding 
exterior parabolic fegment ; and the interior parabolic 
fegment is to the circumfcribed parallelogram as two to 
three . 

Every thing remaining as in the preceding Prop, and Fig- 81 . 
its Cor. the interior parabolic fegment A c l b i a is 
double the correfponding exterior parabolic fegment 
EAIBlce, and the interior fegment is to .the circum¬ 
fcribed parallelogram r a c s as two to three. 

Part I. If the interior fegment be not double of the 
exterior fegment, it muft either be greater or lefs than 
its double. Firft, let it be greater than the double of 
the exterior fegment, and let the fuperficies o be equal 
to the excels of the interior fegment above the double 
of the exterior. Let correfponding redilineal figures 
be inferibed in the interior and exterior fegments, as in 
the preceding Prop, and let them be fuch that the rec¬ 
tilineal figure a 1 s l c, inferibed in the interior feg¬ 
ment, may be lefs than the fegment in which it is in¬ 
feribed by a fuperficies lefs than o, and let e m n p a e 
be the correfponding redilineal figure inferibed in the 
exterior fegment. Then the redilineal figure aiblc 
is greater than the double of the exterior fegment, and 
therefore its half is greater than the exterior fegment. 

But the redilineal figure e m n p a e is half of a iblc, 
by Prop. XIII. and confequently the re6tilineal figure 
E m n p a e is greater than the parabolic fegment in 
which it is inferibed : which is abfurd. The interior 
(fegment therefore cannot be greater than the double of 

the 



OP THE PARABOLA, 


J 24 

BOOK the exterior fegment. Secondly, let the interior feg¬ 
ment be lefs than the double of the exterior fegment, 
and confequently the exterior fegment greater than 
half the interior fegment. Let the excefs of the exte¬ 
rior fegment above half the interior be equal to the fu- 
perficies o ; and let correfponding rectilineal figures be 
inferibed in each fegment, as in the preceding Prop, 
fo that the reCHIineal figure e m n p q. e inferibed in 
the exterior fegment may be lefs than the fegment in 
which it is inferibed by a fuperficies lefs than o. Then 
the rectilineal figure e m n v a E is greater than half 
the interior fegment, and therefore its double is greater 
than the interior fegment. Let A ibLca,emnpq e 
be the correfponding redtilineal figures inferibed in the 
two fegments; and then, by Prop. XIII. as ai'blca 
is double of e m n p a e, it is greater than the interior 
fegment in which it is inferibed : which is abfurd. 
The interior fegment is therefore not lefs than the 
double of the exterior fegment. Confequently as the 
interior fegment is neither greater nor lefs than the 
double of the exterior, it is equal to the double of the 
exterior fegment. 

Part II. As, by Part I. the interior fegment is dou¬ 
ble of the exterior, the interior fegment is to the exte¬ 
rior as two to one, and therefore (18. v.) the interior 
fegment is to the triangle e a c as two to three. Con¬ 
fequently, by Cor. Prop. XIII. the interior fegment is 
to the circumfcribed parallelogram racs as two to 
three *. 


* Archimedes was the firft who proved that a parabolic fegment is 
equal to two thirds of the circumfcribed parallelogram. Of this truth 
he gave two dernonftratipns. The firll may be confidered as mechani¬ 
cal, as it depends upon the primary properties of the lever. The fe- 
cond is ftri&ly geometrical, and may be eafily underftood from the 
above; for he proves, that the triangle a p. n is equal to four times 

the 



OF THE ASYMPTOTES OF THE HYPERBOLA, 


125 


DEFINITIONS. 

XIII. 

If a m n be the vertical plane to the oppofite hyper¬ 
bolas e y f, g l h, as in the eighteenth Definition of 
the firft Book, and cut the cone in the fides a m, a n, 
and if planes a k, a i, touching the cone in the fides 
A N, am, cut the plane of the hyperbolas in the ftraight 
lines k a, 1 r, the ftraight lines k a, 1 r are called 
the Ajymptotes of either of the hyperbolas, or of the op¬ 
pofite hyperbolas. 

Cor. 1. As the vertical plane a m n is parallel to the 
plane of the hyperbolas evf, g l h, the afymptotes 
k a, 1 r (16. xi.) are parallel to a n, a m, fides of the 
cone. 

Cor. 2 . The afymptotes k q, i r do not meet the 
curve of either of the oppofite hyperbolas. For the 
planes a k, a i touch the oppofite cones in the fides 
a n, a m, and as the afymptotes are parallel to thefe 
fides, they do not meet either of the oppofite conical 
fuperficies. The Cor. is therefore evident. 

Cor. 3. Any two ftraight lines t s, p o drawn in the 
planes ak,ai from the afymptotes to the fides a n, 
a m, and parallel to the bafe of the cone, are equal, 
and touch the conical fuperficies. For let n m, i k, 
n k, m 1 be the lines of common fe&ion of the bafe 
with the vertical plane, the plane of the hyperbolas, 


the triangle aib, each of thefe triangles being infcribed according to 
the conditions Hated above. The remainder of his demonftration is then 
equivalent to this, (fee Maclaurin’s Algebra, §. 68.) that the fum of 
the infinite feries 1 + ^ &c. can neither be greater or 

lefs than *, the triangle a b c being analagous to x. 

The above Propofition and the foregoing remarks being clearly 
comprehended, the 5th Cor. to Lemma XI. of Sir Ifaac Newton’$ 
DoFtrine of Prime and Ultimate Ratios will be eafily underftood. 

^nd 


BOOK 

Ill. 


Fig. 92. 


r 



J z6 


BOOK 

III. 


Fig. 93' 


OF THE ASYMPTOTES OF THE 

and the planes A k, a i. Then, as a k, s n are paral¬ 
lel, and as s t, being parallel to the bafe, is parallel to 
N k, the flraight line s t (34. i.) is equal to n k. For 
the fame reafons o p is equal to m i. Alfo n k, m i 
are equal. For if n k, m i be parallel, then w k is a 
parallelogram, and (34. i.) n k is equal to m 1. But 
if n k be not parallel to m i, let them meet in w; and 
as they are in the tangent planes, they will touch the 
circle m d n, the bafe of the cone. The flraight lines 
w n, w m (36. iii.) are therefore equal, and (2. vi.) 
w n : n k : : w m : m 1. Confequently (14. v.) n k 
is equal to m i, and therefore st,op are equal, and 
as they are in the planes a k, a 1, they touch the co¬ 
nical fuperficies. 

XIV. 

The angle 1 c K, or q c r, within which either of 
the oppofite hyperbolas is fituated, is called the interior 
angle of the afymptotes ; and the angle rck, oraci, 
adjacent to it, is called the exterior angle of the ajymp- 
totes. 



PROP. XV. 

The point in which the afymptotes of an hyperbola cut one 
another is the center of the hyperbola ; and any flraight 
line pafjing through the center and falling within the 
interior angle of the afymptotes is a trarfverfe diame¬ 
ter ; but any flraight line paffng through the center and 
falling within the exterior angle of the afymptotes is a 
fecond diameter of the hyperbola. 

Part I. Let a e d, e f g be oppofite hyperbolas, and 
let their afymptotes 1 p, r n cut one another in c ; c 
is the center of the hyperbola, or oppofite hyperbolas. 

In the curve of the hyperbola a b d take any two 
points a, d, and draw the flraight line A D. Then a d 

will 



r/ahXM /’tu/c nU 































































✓ 





ft 





« 










* 


























• * 









I 




\ 










I 




t)F THE ASYMPTOTES OF THE HYPERBOLA* 

will meet both the afymptotes ; for if it were parallel 
to either of the two, it would meet the curve in one 
point only, by Cor. i. to the thirteenth Definition, and 
Prop. XIV. Book I. Let a d meet the afymptote r n 
in n, and the other in p; and let l m, parallel to a d, 
touch the hyperbola abd in b, and meet r n in l, and 
i p in m. Let n v, l x, m y, p w be ftraight lines pa¬ 
rallel to the bafe of the cone in which the hyperbola 
was formed, and let them be fuppofed to have touched 
the conical fuperficies in the points v, x, y, w ; and 
then, by Cor. 3. to the thirteenth Definition, n v, 
l x, m y, p w are equal. Alfo, by Cor. 1. Prop. XL 
and Cor. 1. Prop. X. Book I. b m 2 : my 2 : : bl 2 : lx 1 ; 
and A p X p d : p w 2 : : D n X N a : N v 2 . Confe- 
quently (14. v.) b m 2 is equal to b l 2 , and a p x p d 
is equal to d n x n a ; and therefore bm is equal to 
B l, and, by Lemma VI. p d is equal to na. Draw 
c b, and let it meet a d in o ; and then (4. vi.) c b : 
c o : : b l : o n : : b m : op, and therefore (14. v.) 
o n is equal to op. Confequently c b bife&s a d, 
and for the fame reafons it will bife£t any ftraight line 
parallel to ad in the hyperbola; and therefore, by 
Cor. i. Prop. III. Book II. c B is a diameter. Next, 
let the point h be in the curve of one hyperbola, and 
v in the other. Draw f h, and let it meet r n in a, 
and 1 p in m. Let m y, a z be two ftraight lines pa¬ 
rallel to the bafe of the cone in which the fe&ions 
were formed, and let them be fuppofed to have touched 
the conical fuperficies in y and z. Then, by Cor. r. 
Prop. X. Book I. h m x m f : m y 2 : : h a X q f : 
a z 2 , and therefore, fty Cor. 3. to the thirteenth Defi¬ 
nition (and 14. v.) h m x m f is equal to h q x a f, 
and, by Lemma VI. m h is equal to q f. Confequent¬ 
ly if c s be drawn bifecding o, m in s, it will bife& 
F h j and in the fame manner, as above, it may be 

proved. 


*27 

00 K 
111 . 






128 


OP THE ASYMPTOTES OF THE HYPERBOLA: 


BOOK proved, that c s will bifeft any other ftraight line g K 
IIL parallel to f h, and terminated by the oppofite curves. 
By Cor. i. Prop. III. Book II. c s is therefore a diame¬ 
ter, and confequently c is the center of the hyperbola. 

Fig. 92. Part II. Every thing remaining as in the thirteenth 
Definition, any ftraight line l v palling through the 
center c, and falling within the interior angle icxof 
' the afymptotes, is a tranfverfe diameter of the hyper¬ 
bola. 

For draw a c, and through Ac, l c v let a plane 
pafs, and let its line of common fe&ion with the ver¬ 
tical plane be a z. Then (16. xi.) a z, c v are paral¬ 
lel, as are alfo an, c k; and therefore (10. xi.) the 
angles z a n, v c k are equal. In the fame manner it 
may be demonftrated that the angles m a z, 1 c v are 
equal, and that the angle M A n is equal to the angle 
ick. The ftraight line A z therefore, palling through 
a the vertex of the cone, falls within the oppofite fu- 
perficies, and confequently, as in Part II. Prop. IX. 
Book I. it may be proved, that Lev meets the oppo- 
iite conical fuperficies. The ftraight line lcv there¬ 
fore, palling through c the center, meets the curves of 
the oppofite hyperbolas, and confequently is a tranf¬ 
verfe diameter. 

Fig. 93. Part III. Any ftraight line c s, palling through c 
and falling within r c p, the exterior angle of the 
afymptotes, is a fecond diameter of the hyperbola. 

For if through o, any point within the hyperbola 
a b d, a ftraight line as a d be drawn parallel to c s, 
it will meet the afymptotes, by Lemma III. and there¬ 
fore, as it mull cut the curve in two points, by Cor. 1. 
Prop. II. Book II. it is a fecond diameter. 

Fig. 93. Cor. 1. If a ftraight line as l m touch an hyperbola 
and meet the afymptotes, its fegments between the 
point of contact and the afymptotes will be equal. 

And 



12 9 


OP THE ASYMPTOTES OF THE HYPERBOLA, 

And if a ftraight line cutting an hyperbola,, or oppofite 
hyperbolas, meet the afymptotes^ its fegments between 
the curve or curves and the afymptotes will be equal. 
This is evident from the above; for it was proved that 
N A is equal to p d, and h m to f q. 

Cor . 2. If through any point, as p, in either afymp- 
tote, a ftraight line, as k g, be drawn, meeting the op¬ 
pofite hyperbolas in K and g ; and a ftraight line, as 
p A, be drawn, meeting the curve of the hyperbola 
a b d in d and A ; the redlangle under k p, p g will 
be equal to the fquare of the femidiameter parallel to 
K. g, and the rectangle under a p, p d will be equal to 
the fquare of the femidiameter parallel to a p. For let 
e b be the diameter parallel to k g, and, the reft re¬ 
maining as above, let the ftraight line c u be parallel 
to the bafe of the cone in which the fedtion was 
formed, and let it be luppofed to have touched the 
'conical fuperficies in u. Then, by Cor. 3. to the thir¬ 
teenth Definition, p w, c u are equal, and, by Prop. 
XII. Book I. k p X p G : p W 2 : : B c X c e or c B 3 : 
c v 2 ; and therefore, as p w 2 , c u 2 are equal, k p x p g 
is equal to c b\ Again, by Prop. V. Book II. kp x 
p g is to a p x p d as c b 2 to the fquare of the femidi¬ 
ameter parallel to ap; and therefore (14. v.) the Cor* 
is evident. 

Cor. 3. A ftraight line as l m, touching the hyper¬ 
bola a b d in b, and meeting the afymptotes in l, m, 
is parallel and equal to the fecond diameter conjugate 
to the tranfverfe diameter e b, paffing through the 
point of contact. For it may be proved, as in the laft 
Cor. that the fquare of b m is equal to the fquare of 
the femidiameter parallel to it. Confequently, by Cor. 
1. preceding, and by Cor. 3. Prop* III. Book II. this 
Cor. is evident. 

jc PROP, 


OO K 
III. 



13 ° 


BOOK 

III. 


Fig. 94. 




OF THE ASYMPTOTES OF THE HYPERBOLA. 

PROP. XVI. 

According as a tranfverfe diameter of an hyperbola is 
greater, equal to, or lefs than its conjugate diameter, 
the interior angle of the ajymptotes is an acute, a right, 
or an obtufe angle 5 and any other tranfverfe diameter 
is greater, equal to, or lefs than its conjugate diameter . 

Let g b be an hyperbola, of which c is the center, 
and ck,ci the afymptotes, and let a b be any tranf¬ 
verfe diameter, and d e the diameter conjugate to it; 
according as a b is greater, equal to, or lefs than d e, 
the interior angle kci of the afymptotes is an acute, 
a right, or an obtufe angle; and any other tranfverfe 
diameter f g is greater, equal to, or lefs than h l the 
diameter conjugate to it. 

For let n 1, touching the hyperbola in b, the vertex 
of a b, meet the afymptotes in n, 1 ; and let K m,| 
touching the hyperbola in g, the vertex of f g, meet 
the afymptotes in k, m. Then, by Cor. 1. and Cor. 2.1 
Prop. XV. n 1 is bife&ed in e, and km in g; and N 1 
is equal to d e, and k m equal to h l, and therefore,,' 
according as a b is greater, equal to, or lefs than d e, 
c b is greater, equal to, or lefs than b r. But if with 
B as a center and b 1 as a diftance a circle be defcribed, 
its circumference will pafs through n, and according 
as c b is greater, equal to, or lefs than B 1, it will pafs 
between c and b, through c, or on the oppofite fide of 
c from b. Confequently (by 31. iii. and 21. i.) ac¬ 
cording as c b is greater, equal to, or lefs than b i, the 
angle kci is an acute, a right, or an obtufe angle. 
Again, if with g as a center and g m as a difiance a 
circle be defcribed, its circumference will pafs through 
K, and (by 31. iii. and 21. i.) according as the angle 
K c 1 is an acute, a right, or an obtufe angle, the cir¬ 
cumference of the circle will pafs between c and g, 

through 




OF THE ASYMPTOTES OF THE HYPERBOLA. 

through c, or on the oppofite fide of c from g. Con- B o 

lequently, according as k c i is an acute, a right, or an_ 

obtufe angle, cg is greater, equal to, or lefs than gm, 

The Propofition therefore is evident. 

If two conjugate diameters of an hyperbola be equal, 
or if the angle contained by the afymptotes be a right 
one, it is called an Equilateral Hyperbola . 

PROP. XVII. 

The re&angle under two Jlraight lines drawn from a point 
in the curve of an hyperbola to the afymptotes is equal to 
the reEtangle under two Jlraight lines, parallel to them , 
drawn from any other point in the curve of the J'ame or 
oppofite hyperbola to the afymptotes . 

The re&angle under the two ftraight lines a e, a d. Fig, 
drawn from the point A in the curve of the hyperbola 
a v to the afymptotes c h, c k, is equal to the re$> 
angle under the ftraight lines b f, b g, parallel to a e, 
a d, drawn from the point b in the curve of the fame 
or oppofite hyperbola to the afymptotes. 

For draw A b, and let it meet the afymptotes in h 
and k. Then, as a e, b f are parallel, (4. vi.) a e : 

B f : : h a : h b ; and as b g, a d are parallel, b g : 

A d : : k b : k a. But, by Cor. 1. Prop. XV. ha is 
equal to k b, and therefore h b is equal to k a. Con- 
fequently (it. v.) a e : b f : : b g : a d*, and (16. vi.) 

A E X a d is equal to bf x b g. 

Cor. 1. If a e, b f be parallel to the afymptote c kJ 
and A d, b g be parallel to the afymptote c h ; the pa- 

* This is the property referred to by-writers on Natural Philofophy, 
when they prove, that the curve formed by the upper furface of a li¬ 
quid raifed by the force of attraction between two plates, meeting at 
one end, and kept at a {mail diftance from one another at the other, is 
an hyperbola. 

K 2 


rallelograms 



1$Z 


OF THE ASYMPTOTES OF THE HYPERBOLA. 


BOOK rallelograms ed,fg (14. vi.)are equal. For the angle 
f 1IL at c being common to the two parallelograms, they are 
equiangular, and, by the above, the fides round the 
equal angles are reciprocally proportional. 

Cor . 2. If from any two points as A, B in the curve 
of an hyperbola avb two ftraight lines a e, bf, pa¬ 
rallel to one of the afymptotes as c k, be drawn to the 
other afymptote c h ; then cf : c e : :ea : fb. And 
the femitranfverfe diameters c a, c b being drawn, the 
triangles cfb,cea are equal. 

Cor. 3. If c h, c k be the afymptotes of an hyper¬ 
bola A y b, and if from any two points f, e in c h, 
ftraight lines f b, e a be drawn parallel to c k, and if 
f b be drawn to the curve and e a towards it, and if 
e c be to c f as f b to e a, the point a muft alfo be in 
the curve. 


DEFINITIONS. 

XV. 

Fig. $6. If from c the center of the hyperbola m n g any two 
femitranfverfe diameters c n, c q, be drawn to the 
curve, the figure c n a bounded by the femidiameters, 
and the curve n a is called a Hyperbolic Seblor. 

XVI. 

If c r, c g be the afymptotes of the hyperbola m n a, 
and from any two points n, a in the curve, ftraight lines 
N b, a a parallel to the afymptote c r be drawn to 
the other afymptote c g, the -figure a b n a, bounded 
by the ftraight lines n b, b a, a a and the curve n q, 
is called a Hyperbolic Trapezium. 

Cor. A hyperbolic feCtor c n a and trapezium a e n q 
upon the fame curve are equal. For let c n cut a q in 
P, and then, as by Cor. 2. Prop. XVII. the triangles 
c A g, c b n are equal, the rectilineal trapezium abnp 
1$ equal to the triangle cpa, To thefe equals add 

the 



1‘ltt/f XIV/ku/i /, )-> 



Fifj ./) ( l 


1 V c \ 

V F 

o / 











































































OF HYPERBOLIC SECTORS AND TRAPEZIA. I33 

the figure p n a, and then the hyperbolic fe£tor c n a BOOK 
is equal to the hyperbolic trapezium a b n a. * 

XVII. 

Any fegment as c a, intercepted between c the cen¬ 
ter and a a point in either afymptote, is called an 
Afymptotic Segment , and the point a is called its extre¬ 
mity. 

XVIII. 

Any ftraight line in the plane of an hyperbola, or 
oppofite hyperbolas, parallel to either of the afymptotes 
is called an Ajymptotic Secant . 

PROP. XVIII. 

If from the points in which a ftraight line cuts, and the 
point in which a ftraight line parallel to it touches, an 
hyperbola, ftraight lines parallel to one of the afymptotes 
be drawn to the other, they will cut ojffrom the center 
proportional afymptotic J'egments : and, on the contrary 9 
if from the extremities of three proportional afymptotic 
J'egments ftraight lines parallel to the other ajymptote he 
drawn to the curve of the hyperbola, the ftraight line 
Joining the extreme points in the curve will he parallel 
to the tangent puffing through the middle point in the 
curve . 

Let the ftraight line m a cut the hyperbola m n a Fig. 96. 
in the points m, a, and let the ftraight line g r, pa¬ 
rallel to it, touch the hyperbola in n, and from gl, n, 
m let the ftraight lines a a, n b, m d, parallel to the 
afymptote c r, be drawn to the afymptote c g; the 
afymptotic fegments c a, c b, c d are proportionals. 

On the contrary, if c A, cb, cd be proportional afymp¬ 
totic fegments, and ftraight lines A a, b n, dm, pa¬ 
rallel to the afymptote c r, be drawn to meet the 
curve of the hyperbola in a, n, m, the ftraight line 
k 3 m a. 



134 

BOO 

III. 


OP HYPERBOLIC SECTORS AND TRAPEZIA. 

M a, joining the extreme points, is parallel to G R 
touching the hyperbola in the middle point n. 

Part I. Let the fecant m a meet the afymptotes cg, 
c r in the points h, k, and let the tangent g r, pa¬ 
rallel to the fecant, meet them in g, r. Then, as dm, 
A a are parallel, (io. vi.) h m : h d : : k q : c A, and 
therefore, by Cor. i. Prop. XV. (and 14. v.) c A is 
equal to dh. Again, as by Cor. 1. Prop. XV. g n, 
n r are equal, and as bn, ce are parallel, (2. vi. and 
14. v.) c b, B G are equal. Confequently c A : c b : : 
D h : b g ; and (4. vi.) n H : b G : : D m : b n, and 
therefore (11. v.) c A : c b : : d m : b n. But, by Cor. 
2. Prop. XVII. dm :bn ::cb:cdj and therefore 
(11. v.) c A : c B : : C B : c D. 

Part II. Upon the fecond hypothelis let m a meet 
the afymptotes in h, k, and let the tangent palling 
through n meet them in g, r. Then, as above, it may 
be demonftrated that ca is equal to dh, and c b equal 
toBG ; and therefore ca:cb::dh:bg. But, by 
hypothelis, ca:cb::cb:cd; and, by Cor. 2. 
Prop. XVII. c B : c d : : d m : b N. Confequently, 
(] 1. v.) d H : b G ; : dm : b n, and by alteration d h : 
dm : : b G : b n 5 and therefore as dm, bn are pa¬ 
rallel, the angles (6. vi.) d h m, b g n are equal, and 
(29. i.) h k, g r are parallel. 

Cor. If two parallel llraight lines M q, l f cut an 
hyperbola l m q f, and from the points f, q, m, l 
llraight lines f e, a a, m d, l o parallel to the afymp- 
tote c t be drawn to the other afymptote c s, the 
afymptotic fegments c e, c a, c d, c o will be propor¬ 
tional. On the contrary, if the afymptotic fegments 
c e, c A, c d, c o be proportional, and e f, a a, d m, 
l o, parallel to the afymptote c t, be drawn to the 
curve, the llraight line l f joining t.he extreme points 
Will be parallel to m a joining the mean points. For, 

firft. 



OF HYPERBOLIC SECTORS AND TRAPEZIA. '*35 

firft, if G r, parallel to m a or lf, touch the hyperbola BOOK 
in x and n b, parallel to the afymptote c T, be drawn to _____ 
the other afymptote c s, then by the above (and 17. vi.) 
c a x c d is equal to c b 2 , and alfo c e x c o is equal 
to cb 2 . Confequently c e x co is equal to caxcd, 
and ce:ca::Cd:co. 

Upon the fecond hypothefis let cb be a mean pro¬ 
portional between ca, c d, and let bn 3 parallel to ct, 
be drawn to the curve, and let g r touch the hyper¬ 
bola in n. Then, by the fecond Part of the above, 

G R, m a are parallel. Again, as by hypothefis c e : 
ca::cd:co, CEXCois equal to c A x c D. But 
c e being a mean proportional between c A, c d, cb 2 
is equal to c A xcd, and therefore c e x c o is equal 
to cb 2 , and cb is a mean proportional between c e, 
c o. Confequently as before l f is parallel to G R, 
and therefore (30. i.) m a, lf are parallel. 

PROP. XIX. 

If from the extremities of four proportional afymptotiC'feg - 
merits afymptotic fecants he drawn to the curve of the 
hyperbola , the hyperbolic trapezium between the firft and 
fecond fecant will be equal to the hyperbolic trapezium 
between the third andfourth. And if from the extre¬ 
mities of a feries of afymptotic fegments , in geometrical 
progreffion , afymptotic fecants be drawn to the curve of 
the hyperbola , the hyperbolic trapezia between the firfl 
and fecond fecant , the firjl and third , the firfl and 
fourth , andfo on , will be in arithmetical progrefjion . 

Part I. Let m n q, be an hyperbola, of which c is Fig. 96 . 
the center and c t, c s the afymptotes, and in c s let 
c e be to c a as c d to c o, and let e f, a a, D m, o l 
be afymptotic fecants drawn to the curve, the hyper- 
K 4 bolic 



13^ OF HYPERBOLIC SECTORS AND TRAPEZIA. 

B n? K ^°^* 1C tra pkzium e A a F is equal to the hyperbolic tra- 
_ pezium D O L M. 

For m a, l r being drawn, they will be parallel, by 
Cor. Prop. XVIII. Draw c l, c m, c a, c f. Let c v 
be the diameter to which the parallels m q, jl f are 
double ordinates, and let it meet m a in t, and l f in 
v. Then (38. i.) the triangle c l v is equal to the tri¬ 
angle c f v, arid the triangle c m t is e qual to the tri¬ 
angle c a T; and as 

lines parallel to m q and terminated by the curve, the 
fpace T m l v is equal to the fpace T d pv. Confe- 
quently (axiom 3 . i.) the hyperbolic fe&or c v a is 
equal .to the hyperbolic lector c m l ; and therefore, 
by Cor. to the Sixteenth Definition, the hyperbolic tra¬ 
pezia e a a f, d o l m are equal. 

Part II. The reft remaining as above, lei c a, c b, 
c d, c x, &c. be a feries of afymptotic fegments in geo¬ 
metrical progrefiion, and let the afymptotic fecants 
a a, bn, dm, x y, 8 c c. be drawn to the curve ; the 
hyperbolic trapezia a b n q, a d m q, a x y q, &c. are 
in arithmetical progrefiion. 

For let g r touch the hyperbola in n, and then, by 
Prop. XVIII. it is parallel to m a. Let the diameter 
c t pafs through n, and then, by Prop. II. it bife£Is 
jvi a in t ; and (38. i.) the triangles c T a, c t m are 
equal. And as c T bife&s every ftraight line parallel 
to m a, and terminated by the curve, the fpace n m t 
is equal to the fpace Nax, Confequently (axiom 3 . i.) 
the hyperbolic fe&ors c a n, c n m are equal ; and 
therefore, by Cor. to the ftxteenth Definition, the hy¬ 
perbolic trapezia a b n a, b d m n are equal. As, by 
hypothecs, c n is to c D as c d to c x, it may be 
proved, in the &me way, that the hyperbolic trapezia 
r; d m n, i) x y m are equal ; and the fame 'mode of 
proof may be extended to any number of terms, Con¬ 
fequently 




OP HYPERBOLIC SECTORS AND TRAPEZIA, I37 

fequently the hyperbolic trapezia abnq, a d m a, book 
axyQj &c. are in arithmetical progreffion. ]I1, 

SCHOLIUM. 

As the hyperbola and its afymptotes may be inde¬ 
finitely extended, it is evident that a feries of afymp- 
totic fegments in geometrical progreffion, and a corre- 
fponding feries of hyperbolic trapezia in arithmetical 
progreffion, may be continued to any affigned number 
of terms. From the nature of logarithms, therefore, 
the feries of afymptotic fegments ca,cb,cd, &c. as 
above, is analogous to a feries of natural numbers in 
geometrical progreffion, and the feries of hyperbolic 
trapezia a b n a, A D M a, &c. as above, is analogous 
to the logarithms of thefe natural numbers. To enter 
into an explanation of thefe analogies would be incom¬ 
patible with the defign of this work. The reader may 
find full information on the fubje& of logarithms in 
the volumes entitled, u Scriptores Logarithmici/' pub- 
lifhed by Francis Mafcres, Efq. F. R. S. Curfitor Raron 
of the Court of Exchequer. To this Gentleman the 
mathematical world are highly indebted for his perfe- 
vering exertions and liberality in the caufe of fcience. 

He has employed his great abilities in endeavours to 
render fome of the moft important fubjecls perfpicuous, 
and he has expended large fums in the publication of 
fcarce mathematical tracts, and made prefents of many 
copies of them, with the highly laudable motive of 
promoting learning and diffeminating knowledge. 

Within thefe twenty years, laft paft, much has been 
done in this country to facilitate the application of lo¬ 
garithms, and to extend their utility. In 1785 Dr. 

Hutton of Woolwich publilhed in 8vo. extendve Ta¬ 
bles of them, to which he prefixed u A large and ori¬ 
ginal Hiffory of the Difcoveries and Writings relating 

to 



J 3 8 

BOOK 

lit. 


Fig. 97. 


OF HYPERBOLIC SECTORS AND TRAPEZIA. 

to thofe fubje&s.” Thefe Tables are judicioufly ar¬ 
ranged, and are very valuable for general ufe. The 
hidory prefixed to them is a mafterly performance; it 
gave rife to the publication mentioned above, entitled, 
(C Scriptores Logarithmici.” 

In the year 1792 a quarto volume was publiflied, 
under the patronage of the Board of Longitude, en¬ 
titled, “ Tables of Logarithms of all Numbers, from 1 
to 101000; and of the Sines and Tangents to every fe- 
eond of the Quadrant. By Michael Taylor, Author of 
the Sexagefimal Table.” As Mr. Taylor died before 
the Tables were entirely printed, the Rev. Dr. Malke- 
lyne, Adronomer Royal, fuperintended their comple¬ 
tion. He alfo wrote the Preface and Precepts for the 
ufe of the Tables; and thefe he executed with that 
care, learning, and ability, for which he is fo juftly ce¬ 
lebrated in every part of the world where either the 
theory, or pra&ical utility, of Aftronomy is underftood. 
As Taylor’s Tables are accurate, and more extenfive 
than any other extant, the volume is an excellent re- 
foiirce to thofe who aim at a fuperior degree of preci- 
fion in their calculations. 

In a fmall quarto volume of mathematical Efifays, 
publifiied in 1788 by the Rev. John Hellins, (now 
Vicar of Potters’ Pury, Northampton (hire, and F.R. S.) 
there are two Efifays on Logarithms, which difplay an 
intimate knowledge of the fubjedt. A fcientific reader 
will find much gratification in the perufal of thefe 
ElTays. 


DEFINITION. 

XIX. 

If a b be a tranfverfe diameter of the oppofite hy¬ 
perbolas a, b, and d e the fecond diameter conjugate 
to it, and if d e be a tranfverfe diameter of the oppo¬ 
fite 



Op conjugate hyperbolas. 


*39 

fite hyperbolas d, e, and a b the fecond diameter con- book 
jugate to it; the hyperbolas d, e are called the Conju - IIL 
gate Hyperbolas to one or both of the oppofite hyper¬ 
bolas a, b, and, on the contrary, the hyperbolas a, b 
are called the Conjugate Hyperbolas to one or both of 
the oppofite hyperbolas d, e. When all the four hy¬ 
perbolas a, d, b, e are mentioned together, they are 
called Conjugate Hyperbolas. 

Cor. If the diameters a b, d e cut one another in c, 
it is evident that c is the common center of the con¬ 
jugate hyperbolas. 


PROP. XX. 

One of the afymptotes of an hyperbola is parallel to , and 
the other bifcEis , a Jlraight line joining the vertices of 
any two conjugate diameters : and the vertices of fecond 
diameters of an hyperbola are in the curves of the hyper¬ 
bolas conjugate to it. 

Part I. Let ab, d e be any two conjugate diame- Figh¬ 
ters of thehyperbola ah, and let cf,cg be its afymp¬ 
totes, c being the center; one of the afymptotes, as 
c f, is parallel to a e the ftraight line joining the ver¬ 
tices a, e, and the other afymptote c g bifedfs A e. 

For let f g touch the hyperbola ha in a and meet 
the afymptotes in f and g . Then, by Cor. 3. and 1. 

Prop. XV. f g is equal and parallel to d e, and c e is 
equal to fa and alfo to a g. Confequently (33. i.) 
the afymptote c f is parallel to a e. Alfo the angle 
(29. i.) cel is equal to the angle lag, and the an¬ 
gle e c l to the angle agl, and therefore (2 6 . i.) A l 
is equal to e l, and a e is bife£ted by the afymptote 
c g. 

Part II. Let k h be any tranfverfe diameter of t\ie 
oppofite hyperbolas a h, b k, and let m n be the fe¬ 
cond 


*40 


OF CONJUGATE HYPERBOLAS. 


90 0 K 
III. 


T»g. 98. 
99 - 


cond diameter conjugate to it; the vertices m, n of 
this fecond diameter are in the curves of the hyperbo¬ 
las conjugate to the hyperbolas ah, bk. 

For let d, e be the hyperbolas conjugate to A h, 
b k , and let a b, d e be the conjugate diameters com¬ 
mon to the conjugate hyperbolas, as in the nineteenth 
Definition, and c the center. Let f c, gc be the 
afymptotes of the oppofite hyperbolas a, b. Draw b e , 
k n, and let them meet the afymptote f c in p and a ; 
and then, by Part I. bk, kn are parallel to the afymp¬ 
tote g c, and they are bifected by the afymptote f c in 
p and q. Confequently, by Cor. 2. Prop. XVII. c p : 
c a : : k a : b p ; and therefore, on account of the 
equals, cp : cg-: : qn: pe } and as e is in the curve 
of the hyperbola e, n mud be in the curve of the fame 
hyperbola e, by Cor. 3. Prop. XVII. 

Cor. A ftraight line parallel to one of the afymp¬ 
totes, and terminated by the curves of the conjugate 
hyperbolas, is bife&ed by the other afymptote. 

PROP. XXI. 

A quadrilateral fgure, whofe Jldes pafs through the ver¬ 
tices of any tiro conjugate diameters of an ellipfe, or 
conjugate hyperbolas, and touch the ellipfe or hyper¬ 
bolas, is a parallelogram, and is equal to the redangle 
under the axes of the ellipfe or hyperbolas . 

I<et m s l r be a quadrilateral figure, whofe fides 
m s, s l, l r, r m pafs through f, g, h, k, the ver¬ 
tices of the conjugate diameters f h, g k of the ellipfe 
f g h k, or the conjugate hyperbolas f, g, h, k, and 
in thefe points touch the curve or curves; the figure 
m s l r is a parallelogram, and is equal to the re 6 t- 
angle under the axes ab,de of the ellipfe or hyper¬ 
bolas. 


For, 



OF CONJUGATE HYPERBOLAS. 


For, by Cor. 2. Prop. IV. Book II. the tangents book 
M s, r l are parallel to the diameter k g, and the tan- nI ‘ 
gents m r, l s are parallel to the diameter f h. The 
quadrilateral figure mslr is therefore a parallelo¬ 
gram. 

Again, let c be the center, and let c 1 be perpendi¬ 
cular to the tangent s l. Then as f h is bife&ed in c, 
the parallelogram g h is a fourth part of the parallelo¬ 
gram l m ; and, by Cor. 1. Prop. XIX. Book II. c 1 : 
c b : : c d : c h, and c 1 X c h is equal to c b X c d. 

But (35. i.) c 1 x c h is equal to the parallelogram 
g h, and c b x c d is a fourth part of the re&angle 
under the axes A b, d e. Confequently the parallelo¬ 
gram l m is equal to the re&angle under the axes a b, 

D E. 

Cor. All parallelograms contained under tangents, 
palling through vertices of conjugate diameters of an 
ellipfe, or conjugate hyperbolas, are equal to one ano¬ 
ther, and each of them is equal to the re&angle under 
the axes. 

The twelfth Lemma of Sir Ifaac Newton’s Principia, 

Lib. I. and the tenth Propolition, depending upon it, 
are evident from the above. 

PROP. XXII. 

If through a point in the curve of an hyperbola two Jlraight 
lines be drawn parallel to the afymptotcs and mcetmg a 
diameter , the fernidiametcr will be a mean proportional 
between the fgmcnis of the diameter between the center 
and the points of concourfe . 

Through the point 1 in the curve of the hyperbola Fig. ico, 
I b let the ftraight lines 1 t, i a be drawn parallel to 
tlie afymptotes c e, c g, and firft let them meet the 
tranfverfe diameter v b in the points T, A, and let c be 

the 



i4£ 

BOOK 

III. 


Fig. ioi. 

103 . 


OF CONJUGATE HYPERBOLAS. 

the center ; the femidiameter c b is a mean propor¬ 
tional between c t, c A. 

For let 1 a meet the afymptote ce in f, and let b h, 
tk be parallel to i f, or to the afymptote c G, and let 
them meet ce in h and k. Then (34. i.) kt,fi 
are equal ; and (4. vi.) c K : c h : : k t or its equal 
fi:hb. But, by Cor. 2. Prop. XVII. f i : h b : : 
c h : c. f, and therefore (11. v.) c k : c h : : c h : c f. 
Confequently, on account of the parallels kt, hb, f a, 
ct:cb::cb:ca. 

Secondly, the reft remaining as above, let the ftraight 
lines 1 t, i a meet the fecond diameter lm in the 
points n, o, and let 1 a meet the curve of the hyper M 
bola p l, conjugate to b 1, in p; and let pa be paral¬ 
lel to c e or 1 n, and meet the diameter l m in q. 
Then, by the above, cq:cl::cl:co. But, by 
Cor. Prop. XX. if, fp are equal, and therefore, as 
r a, 1 n are parallel, c. a is equal to c n. Confe¬ 
quently cn:cl::cl:co. 

Cor . 1. If two ftraight lines t n, t p, touching an 
hyperbola, or oppofite hyperbolas, in n and p, meet 
one another in t, and if a ftraight line t i parallel to 
one of the afymptotes meet the curve in 1, a ftraight 
line 1 A parallel to the other afymptote and meeting 
N p, the ftraight line joining the points of contact, in 
A will bife£t n p in a. This is evident from the above, 
and Prop. VII. Book II. 

Cor . 2. The reft remaining as in the preceding Cor. 
if t 1 produced meet n p in e, t e will be bife&ed in 
1. For let t l parallel to 1 A, or an afymptote, meet 
the curve in l, and then, by Cor. i.la parallel to t i 
will meet n p in a, the point in which n p is bife£ted; 
and tial is a parallelogram. Draw 1 l, and let it 
meet the diameter c b a in G. Then (34. i.) 1 a, t l 
are equal, and (29. i.) the triangles aig,tlg are 

equi- 



TLueXVf>agt 141 






















































OP THE PARABOLA AND HYPERBOLA.' 143 

equiangular; and therefore (26. i.) ig, gl are equal, SCO A 
and t g is equal to ga. The ftraight lines 1 l, n p ^ 
are therefore ordinates to the diameter cba, and con- 
fequently, by Prop. II. they are parallel; and (2. vi.) 
tg:ga::ti:ie. The ftraight line t e is there¬ 
fore bife£ted in 1. 

PROP. XXIII. 

If a diameter of a parabola , or a ftraight line parallel to 
an ajymptote of an hyperbola, meet two tangents and 
the ftraight lines joining the points of contad, the 
fquare of its fegment between the curve and the ftraight 
line joining the points of contad will be equal to the 
redangle under the fegmenls between the curve and tan¬ 
gents . 

If the diameter of the parabola, or the ftraight line 
parallel to an afymptote of an hyperbola, pafs through 
the point in which the tangents meet one another, the 
Propofition is evident from Prop. V. and Cor. 2. Prop. 

XXII. but if not, let the two ftraight lines m n, m o Fig. 103, 
touch the parabola, hyperbola, or oppofite hyperbolas, 
in the points n, o, and let t x a diameter of the para¬ 
bola, or a ftraight line parallel to an afymptote, meet 
the parabola or either hyperbola in e, the tangents in 
a, d, and the ftraight line joining the points of con¬ 
tact in b ; the fquare of e b is equal to the rectangle 
under A E, e d. 

Cafe 1. If the ftraight lines touching the parabola Fig. to?, 
or hyperbola meet one another in m, from the point d 
in which T x meets one of the tangents draw d l pa¬ 
rallel to the other tangent m o, and let it meet the 
curve in p, l, and the ftraight line n o in k. Then, 
by Prop. XVII. Rook I. the fquare of dk is equal to 
the rectangle under l d, dp; and (4. and 22. vi.) 

a B a , 



144 OR THE PARABOLA AND HYPERBOLA* 

BOOK a B 2 : D b* : : A o 2 : d k 2 or l d x D p. But as t X 
IIL is parallel to a fide of the cone in which the fe&ion 
was formed, by Prop. XVI. Book I. a o 2 : ldxdp:: 
Ae:de; and therefore (n. v.) ae:de::ab j : 
d b 2 . Confequently, by Lemma VIII. a e : b e : : 
be:de; and (17. vi.) a e x e d is equal to b e 2 . 

Fig. 105. Cafe 2. If the ftraight lines touching the oppofite 
hyperbolas meet one another in m, from a or d, fup- 
pofe d, draw, the ftraight line d k parallel to the tan¬ 
gent A o, and let it meet n o in k. Through the 
points A, d draw the ftraight lines g h, p l parallel 
to one another, and let them meet the oppofite hyper¬ 
bolas in the points g, h and p, l. Then, by Prop. Vj 
Book II. g a x a h is to a o 2 as the fquare of the fe- 
midiameter parallel to g h to the fquare of the femidi- 
ameter parallel to a o. And, by Cor. 1. Prop. XVIL 
Book I. and Prop. V. Book II. p d x d l is to d k 2 as 
the fquares of the fame femidiameters. Confequently 
(11. v. and alternation) gaxah : pdxdl:: ao 2 : 
d k 2 . But (4. and 22. vi.) A o 2 : d k 2 : : a b 2 : d b 2 | 
and, by Prop. XVI. Book Lgaxah :p.dx d l : : 
A e : d e. Confequently a e : d e : : A b 2 : d b 2 , and 
therefore, by Lemma VIII. ae:be::be:d e, and 
a e x d e is equal to b e 2 . 

Fig. 105. Cafe 3. If the ftraight lines A o, d n touching the 
oppofite hyperbolas be parallel, then the triangles a bo, 
den will be equiangular, and the proportion will be 
A o 2 : D n 2 : i A b 2 : D b 2 ; and in this cafe, by Prop. 
XVI. Book I. a e : d e : : a o 2 : d n 2 . Confequently 
a e : d e : : a b 2 : D b 2 , and, by Lemma VIII. A e : 
be::be: DEj and A e x d e is equal to b e 2 . 

PROP. XXIV. 

If from two given points in the curve of a parabola , or 
. hyperbola , or the curves of oppofte hyperbolas , two 

1 Jlraighl 



OP THE PARABOLA AHD HYPERBOLA. 


*45 


Jlraight lines be inflected to any third point in the curve BOOK 
of the parabola , or in the curve of either of the oppofte 
hyperbolas , and if they meet a diameter of the parabola , 
or a Jlraight line parallel to an afymptote of the hyper¬ 
bola ; the fegments of this laft mentioned line , between 
the infle&ed lines and the point in which it meets the 
• curve of the fehtion , will be to one another in the fame 
ratio , wherever the point may be in the curve to which 
the lines are infleEled . 

Let n, m be two given points in the curve of the Big. 106. 
parabola, hyperbola, or oppofite hyperbolas, and let *°g* 
the ftraight lines n o, m o inflected from them to any 
point o in the curve of the parabola, or in the curve of 
either hyperbola, meet in b, c the ftraight line t x, a 
diameter of the parabola, or parallel to an afymptote of 
the hyperbola, and let t x meet the curve in e ; the 
fegments e b, e c are to one another in the fame ratio, 
wherever the point o may be taken in the curve. * 

For let tangents palling through m, n, o meet t x 
in f, d, a. Draw m N, and let it meet t x in g ; and 
by Prop. XXIII. ed:eg!:eg:ef. Again, by 
Prop. XXIII. e b 2 is equal to a e x e d, and e c* 
is equal to a e x e f. Confequently e b 1 : e c 2 : : 
aexed:aexef::(i. vi.) ed:ef. But, by the 
above, (and Cor. 2. 20. vi.) e d : e f : : e d 2 : e g 2 $ 
and therefore (it. v.) eb 2 : e c 2 : : ed 2 : e g 2 , and 
(22. vi.) eb:ec::ed:eg. But as the points m, 
n are given, or fixed, and as the ftraight line t x is 
given in pofition, the fegments ed, eg remain the 
fame. Confequently, if the point o be moved round 
the curve of the parabola, hyperbola, or oppofite hy¬ 
perbola, the fegments e b, e c of the ftraight line T x, 
between the infle&ed lines and the curve, wil| be to 
one another in the fame ratio, wherever the point o - 
may be in the curve. 

L 


SCHO- 



14 6 


DESCRIPTIONS OF THE SECTIONS. 


BOOK 

III. 


Fig. 109. 
110. 


SCHOLIUM. 

It may be proper in this place to direct the attention 
of the reader to methods of afcertaining certain parti¬ 
culars in a conic fe&ion, fuppofing the curves of the 
feftions to be given, or ftraight lines to be given, for 
the defcription of the curves. Thefe methods, now to 
be defcribed, might have been delivered as Corollaries 
to the Propofitions on which they depend, or they 
might have been put into the form of Problems ; but 
it appeared more advifeable to referve them for a feries 
of articles in a Scholium. A cautious defire againfl in¬ 
terrupting the reader in the acquifition of new truths 
fuggefted this delay. The eafe with which they are 
deduced from the preceding Propofitions, and the im¬ 
portance of the articles themfelves, induced the author 
to think that the following was the moft proper man¬ 
ner of delivering them, and the moft proper place to 
inlert them. 

1. Let the ellipfe adbe, or the oppofite hyperbo¬ 
las a, d b e, be given to find the center. 

In the ellipfe and in either hyperbola draw the two 
parallel ftraight lines d e, f g, and draw a b bife&ing 
d e in u and f g in k. Let a b meet the curve of the 
ellipfe, or the curves of the oppofite hyperbolas, in A 
and e. Bife6t a b in c, and c will be the center, by 
Cor. 1. Prop. III. Book II. 

If only one hyperbola dbe be given, two other 
ftraight lines mult be drawn parallel to one another, 
but not parallel to d e, f g, and a ftraight line being 
drawn bifefting them will be a diameter. Its concourfe 
therefore with A b will determine the center. 

2. The curve of a conic fe&ion and a point in it be¬ 
ing given, let be required to draw a diameter through 
the given point. 

If the fetftion be an ellipfe, or hyperbola, find the 

cen- 



f‘/n(eX\ I.fMuje 14 0 ■ 































































DESCRIPTIONS OP THE SECTIONS. 


*47 


center by the preceding article, and through the cen- BOOK 
ter and the given point draw a diameter. If the fee- m * 
tion be a parabola, find a diameter, by Cor. 3. Prop. II. 
of this Eook, and parallel to it draw a ftraight line 
through the given point; and, by Cor. 1. to the firft 
Definition, this will be the diameter required. 

3. The curve and a diameter a b of a conic fe&ion 
being given, and any point g in the curve befides a 
vertex of the given diameter, let it be required to draw 
a ftraight line from g ordinately applied to the dia¬ 
meter. 

Firft, let the fe&ion a g b d be an ellipfe. Find the Fi £- I0 9 * 
center c by the firft; article, and through it draw the 
diameter g l. Through the vertex l draw the ftraight 
line l f parallel to a b. Then if l f touch the ellipfe, 
ab,lg will be conjugate diameters, by Cor. 2. Prop. 

IV. Book II. and l g will be ordinately applied to 
a b. But if l f do not touch the ellipfe, let it meet 
the curve again in f. Draw g f, and let it meet a b 
in k, and g f will be the ordinate required. For, as 
c k, l f are parallel, (2. vi.) g c : g k : : c l i ic f, 
and therefore (14. v.) g f is bifefted in k. Secondly, Fig. no. 
let the fe&ion f b g be an hyperbola, or parabola. In 
the diameter a b take any point m, and the ftraight 
line g m being drawn, produce it to l, fo that m l 
may be equal to G m. Then if the point l be in the 
curve, g l will be the ordinate required; but if l be 
not in the curve, draw the ftraight line l f parallel to 
a b, and let it meet the curve in f, according to Prop. 

IX. Book I. or Cor. 1. Def. I. of this. Draw g f, and 
it will be the ordinate required. For, if it meet a b in 
k, it may be proved, as above, that g f is bife&ed in k. 

4. The curve of a conic fe6tion fbg being given, Fig. no. 
and a point B being given in it, let it be required to 

draw a ftraight line through b to touch the feclion. 

l 2 Through 



148 


DESCRIPTIONS OP THE SECTIONS* 


book Through b draw A b, a diameter of the fe&ion, andy 
IiL by the lafl article, draw gf an ordinate to it. Through 
the vertex b draw the ftraight line t b parallel to g f, 
and, by Prop. II. t b will be the tangent required. 

Fig. in. 5. Two unequal ftraight lines a b, d e being given, 
bife< 5 ling one another in c at right angles, let it be re¬ 
quired to defcribe the curve of an ellipfe, of which 
A b, d e (hall be the axes, and c the center. 

Let a b be greater than d e, and consequently the 
tranfverfe axis. Find the foci f, o in A b, by Cor. 2* 
Def. XI. Book II. Let the ends of a thread or firings 
equal in length to a b, be fixed to the points f, o. By 
means of the pin p let the thread or firing be ftretched; 
and, while it continues uniformly tenfe, let the end or 
point of the pin p move round in the plane, in which 
a b, d e are fttuated, till it return to the fame place 
from which it began to move. The line traced by the. 
end or point of the pin p is the curve of an ellipfe, as 
is evident from Prop. XIII, Book II. and a b, d e are 
the axes. 

Fig. 112. 6 . Two ftraight lines a b, d e being given, bife£ling 

one another in c at right angles, let it be required to 
defcribe the curve of an hyperbola, of which a b ftiall 
be the tranfverfe and d e the conjugate axes. 

In a b produced both ways let the foci f, o be 
found, by Cor. 3. Def. XI. Book II. Let one end of a 
thread or firing f p r be fixed to the point f, and let 
the other end be fixed to the extremity r of the ruler 
o r, and let the length of the ruler exceed the length 
of the thread or firing by the ftraight line a b. Let 
o, the other extremity of the ruler, be fixed to the 
point o, and let the ruler revolve about o as a center. 
By means of the pin p let the thread or firing be 
ftretched, and let the part between p and r be kept 
clofe to the edge of the ruler ; and while the ruler re¬ 
volves. 



DESCRIPTIONS OP THE SECTIONS. 

volves, and the thread or firing is kept uniformly tenfe, 
Jet the end or point of the pin p trace the line g b h 
in the plane in which a b, d e are fituated. The line 
g b h will be the curve of an hyperbola, of which a b 
is the tranfverfe and d e is the conjugate axes, and c 
is the center, as is evident from Prop. XIII. Book II. 

7. The ftraight line d x, of indefinite length, being 
given, and f being a point given without it, let it be 
required to deferibe the curve of a parabola, of which 
D x (hall be the directrix and f the focus. 

Place the edge of a ruler rdxl along the line d x, 
and keep it fixed in that pofition. Let g y e be a ruler 
of fucli a form that the part g y may Aide along the 
edge d x of the fixed ruler rdxl, and the part y e 
may be always perpendicular toox. Let e p f be a 
thread or ftring of the fame length with the part y k 
of the moving ruler, and let one end of it be fixed to 
the ruler at k, and let the other end be fixed to the 
point f. By means of the pin p let the thread or ftring 
be ftretched, and the part between p and k be kept 
clofe to the edge of the ruler. While the ruler g y e 
Aides along the edge dx of the fixed ruler, and the 
thread or ftring is kept uniformly tenfe, let the end or 
point of the pin p trace the line a p b c on the plane, 
in which the line dx and the point f are fituated. The 
line a p b c will be the curve of a parabola, of which 
d x is the directrix and f the focus, as is evident from 
Cor. 3. Prop. VIII. 

Several writers on Conic Sections have defined the 
ellipfe, hyperbola, and parabola by the defeription in 
Article 5, 6, and 7 refpectively; and from thefe de- 
feriptions, as founded on a primary, they have deduced 
other properties of the fections. 

8 . Two ftraight lines A b, d e being given, bife£t- 
ing one another in c but not at right angles, let it be 

L 3 re- 


M 9 


BOOK 

III. 


Fig. 113. 


Fig. 1 14. 
115. 



i5o 

BOOK 

III. 


rig. 116 . 


DESCRIPTIONS OF THE SECTIONS, 

required to defcribe an ellipfe, or hyperbola, of which 
ABjDE fhall be conjugate diameters, and c the center. 

In c d, produced in the ellipfe but between c and d 
in the hyperbola, take the point n, fo that the red¬ 
angle under c d, d n may be equal to the fquare of 
c b. Through d draw the ftraight line m q parallel 
to A b, and bifed c n in i . Draw i p perpendicular to 
c n, and let it meet m q in p; and then it is evident 
(4. i.) that ftraight lines drawn from p to N and c will 
be equal. With p as a center therefore, and pn or 
p c as a diftance, let the circle m c a n be defcribed, 
and let it meet the ftraight line m q in m and q. 
Draw the ftraight lines a c, m c ; and from d draw 
d k perpendicular to a c, and d h perpendicular to 
m c. In a c take c l, c it each a mean proportional 
between c q, c k ; and in m c take c f and c g each 
a mean proportional between cm, c h. Then will 
r l, g f be the axes of the ellipfe, or hyperbola, pro- 
pofed to be defcribed, as is evident from Cor. 2. Prop. 
IX. (and31. iii.) and Prop. VII. Book II. Confequent- 
ly the foci may be found, and the defcriptions of the 
curves may be completed, as in the 5th and 6th Articles. 

9. The ftraight line g e being given in pofition and 
magnitude, and the ftraight line a b bife&ing it in b, 
let it be required to defcribe a parabola, of which a b 
ftiall be a diameter, and g e a double ordinate to it. 
Let the ftraight line p be a third proportional to a b, 
b g, and produce b a to y, fo that a y may be a fourth 
part of p. Through y draw d x at right angles to 
y b, and through a draw a n parallel to g e. Make 
the angle n a f equal to the angle nay, and make 
a f equal to ay. A parabola defcribed with the focus 
f and the dire&rix d x, as in the 7th Article, will be 
the fe&ion required, as is evident from Prop. II. IX. 
and Cor. 2. and 3. Prop. XI. 


10. The 



DESCRIPTIONS OF THE SECTIONST 

io. The diameter a b of an ellipfe or hyperbola p b, 
and pl an ordinate to it being given, let it be required 
to find the diameter conjugate to a b. 

Let the ftraight line m be a mean proportional (13. 
vi.) between the abfcilfes A f, f b ; and, c being the 
center of the fe&ion, let m be to p l as c b to c d a 
ftraight line parallel top l. Then will c d be half the 
conjugate diameter required. For, by hypothefis, (and 
22. vi.) m 2 : p l 2 : : c B z : c d\ But (17. vi.) m 2 =2 
alxlBj and therefore alXlb:pl j ::cb 2 :cd 2 . 
Confequently CD is the femiconjugate diameter to 
a b, by Prop. IV. and V. and Def. VIII. Book II, 


1. 4 


* 5 * 

BOOK 

III. 


Fig. hi. 

112. 


LEMMAS 







* 

























• t 











' >, j / • iI- i 

' '*** ’ , A . 

' 

• • " ’ 






. ■ :■> *>': : ■ ’ : : ’ • '• ; i • 

*v j»jjV ; ill . , U . 

-.41 lr V h C I 









i. . . . -: •. 






















Piute XVII.page tyi. 



/ Basin’ sc 


(/ _ 

.1 

»P 

l o c 

L. F ) 


A 


A 

r \ T? 


B \ 




\ D 


p 

A 

C 

K 

b! f 



Fig. 114. / 

F\ 





A<^ II 


^P 


\\ K /L / 



A\ 










































































LEMMAS 


FOR 

THE FOURTH BOOK 

OF 

CONIC SECTIONS. 


LEMMA IX,* 

Let it be required to draw a Jlraight line to touch two 
given circles alb, d e m. 

Let c be the center of a l b, and f the center of Fig. iaS. 
d e m, and draw c f. Let c G be the excefs of the 
radius of alb above the radius of d e m ; and with c 
as a center, and c g as a diftance, defcribe the circle 
h g. From the point f draw fh (17. iii.) touching the 
circle h g in h. Draw c h, and let it meet the circum¬ 
ference of alb in A j and draw f d parallel to c a, 
and let it meet the circumference of n e m in d. 

Draw a d, and it will touch the given circles. For 
by the conftru&ion a h, d f are equal and parallel, 
and (16. iii.) a h f is a right angle. Confequently 
(33 • i*) a d f h is a parallelogram, and (Cor. 4 6. i.) 

* The numbering of the Lemmas is continued from thofe prefixed 
to the firft Book. This manner of numbering them was found moil 
convenient for reference. 


the 





154 


LEMMAS FOR THE FOURTH BOOK. 


Fig. *29. 
150. 


the angles at A and d are right ones, and therefore 
(16. iii.) a d touches the circles. 

If it be required to draw a ftraight line to touch the 
circles and cut c f, let c f be fo divided in k that c k 
may be to k f as the radius of the circle a l b to the 
radius of the circle d e m. Draw k l (17. iii.) to 
touch the circle alb in l. Draw c L and F m pa¬ 
rallel to it, and let f m meet l k in m ; and l k will 
touch the circle dem in m. For (15. and 29. i.) the 
triangles c k l, f k m are equiangular, and therefore 
(4. vi.) c K : K F : : c l : f m. Confequently, by 
the conftru&ion, the point m is in the circumference of 
the circle dem, and as the angles f m k, c l k are 
equal, and the angle c l k a right one, L m (16. iii.) 
touches the circle d e m in m. 

LEMMA X. 

If a magnitude A be to a magnitude B as a magnitude c 
to a magnitude d ; then A will be to B as the differ¬ 
ence of the antecedents A, C to the difference of the con - 
feqitents b, d.' 

For let a be greater than c, and confequently (14. v.) 
b greater than d. Then, by alternation, a : c : : b : d, 
and (17. v.) A — c : c : : B ~ d : D ; and again, by 
alternation, c : d : : A — c : b — d. But, by hypo¬ 
thecs, a : b : : c : d, and therefore (IT. v.) a : B : : 
A — c : b — D. 


definitions: 

1. 

If the ftraight line a d be fo divided in the points B, 
c, that the whole line a d be to one of the extreme 
parts a b, as the other extreme part d c to the mid¬ 
dle part b c, the ftraight line a d is faid to be Har¬ 
monically 


LEMMAS FOR THE FOURTH BOOR. 


*55 


monically divided ; and the points A, B, c, d are called 
the Points of harmonical divijion, or Harmonical Points. 

Cor . 3. It is evident that each extreme part is greater 
than the middle part. 

Cor. 2. If the two extreme points A, p, and b one 
of the middle points of an harmonical diviiion be given, 
the other point c may be found (10. vi.) by dividing 
the fegment b d in c, fo that the part c d may be to 
b c as a d to a b. It is evident that no other point 
belides c can be found, which can be a fourth point of 
this diviiion. 

Cor . 3. The two middle points B, c, and A one of Fig. 130. 
the extreme points of an harmonical diviiion being 
given, the other point d may be found. For from the 
point A draw the ftraight line A e, and as A b to B c, 
fo let a e be to the fegment e f, taken towards a. 

Draw f c, and let e d drawn parallel to f c meet a c 
produced in d. Then (2. vi.) ad:dc::ae:ef; 
and therefore by the conftru&ion ad: d c : :ab:bc, 
and confequently d is the other point of the diviiion. 

It is evident that no other point befides d can be 
found, which can be a fourth point of this diviiion. 

For, by converfion, as the excels of a b above b c to 
b c, fo is a c to c B. 

II. 

If the ftraight line a d be divided harmonically in Fig. 129. 
the points a , b, c y d 9 and four liraight lines a e, b e, I3C, ‘ 
c e, d e, any way produced through the points of di¬ 
viiion, be parallel or meet one another in the point e ; 
thefe four liraight lines are called Harmonicals. 

Cor. Every thing remaining as above, any ftraigbt 
line a d, parallel to a d , and meeting the harmonicals in 
a, b, c, d, will be harmonically divided in thefe points. 

For, (29. and 15. i.) on account of the equiangular tri¬ 
angles. 


1 36 LEMMAS FOR THE FOURTH BOOK. 

angles, a d : d e : : A d : d e, and d e : d c : : d e : 
i>c. Confequently, 

a d : d E : d c 
A D : D E : D C, 

and (22. v.) a d : d c : : A d : d c. Again ab ib e : : 

ab ; BE; and b E : b c be : B c 5 and therefore 

a b : b e : b c 

A B : B E : b C. 

Confequently (22. v.) 'ab : b c : : ab : bc; and there¬ 
fore (11. v.) a d : d C : : A b : b c, as, by hypothefis, 
ad i d c : : a b : b c, 

LEMMA XI. 

Fig. 129, The rejl remaining as in the jirjl and fecond Definitions 
I3 °* and their Corollaries , if a Jlraight line G H parallel to 
any one of the harmonicals a E, b E, c E, d e, meet the 
other three , it will be bifefted in the middle point of con - 
courfe. And , on the contrary , \iffour Jlraight lines e r>, 
ec,eb, EA meet one another in e, and if the Jlraight 
line G h parallel to any one of them , and meeting the 
other three , be bifetted in the middle point of concourJe y 
the Jlraight lines E D, E c, E B, EA will be harmo¬ 
nic a Is. 

Tig. 130. Part I. Firft, let the flraight line g h be parallel to 
the harmonical d e, and meet a e, b e, c e in G, b, h ; 
g h is bife&ed in the middle point b. For through b 
draw the flraight line A d parallel to a d, and meeting 
the harmonicals a e, c e, d e in a, c, d. Then the 
flraight line ad is harmonically divided, by the Cor. 
to the fecond Definition, and therefore d a : a b : : 
d c : c B. But (4. vi.) da:ab::de:bg; and 
bc:cb::de:bh. Confequently (11. v.) D E : 
b g : : d e : b h, and therefore (14. v.) G b, b h are 
equal. 


Se- 


LEMMAS FOR 'THE FOURTH BOOX. 157 

Secondly, let the ftraight line gh be parallel to the Fig. 
harmonical c e, and meet a E,h,iE in a, h, g ; it 
will be bifecled in the middle point a. For through 
A draw the ftraight line a d parallel to a d , and meet^ 
ing the harmonicals b e, c e, d e in b, c, d. Then, 
by the Cor. to the fecond Definition, ad : dc: :ab: 
b c. But (4. vi.), ad:dc::ga:ec; and a b : 

B c : : A H : e c. Confequently (11. v.) ga: EC;: 
ah : E c, and therefore (14. v.) ga, ah are equal. 

Part II. Firft, let the ftraight line gh be parallel to Fig. i3«. 
Ed, and meet the ftraight lines e c, eb,ea in h, b, 
g, and let it be bife&ed in the middle point b. Through 
B draw the ftraight line a d, and let it meet the ftraight 
lines e a, e c, e d in a, c, d. Then (4. vi.) de;bg;; 
d a : a b, and de:bh:;DC;cb. Confequently 
(7. and 11. v.) d a : a b : : d c : c B. 

Secondly, let the ftraight line gh be parallel to e c, Fig. 129. 
and meet the ftraight lines e d, e b, e a in the points 
g, h. A, and let it be bife&ed in the middle point a. 

Through a draw the ftraight line a d, meeting the 
ftraight lines e b, e c, e d in b, c, d. Then (4. vi.) 

1 ga:ec::ad:dc; and a ii : e c : : a b : b c. 
Confequently (7. and n.v.j ad : dc : : ab : bc. 

LEMMA XII. 

If four harmonicals meet any ftraight line , the ftraight Tine 

will be harmonically divided in the points of concourfe . 

If the harmonicals are parallel to one another, this is 
evident (from 10. vi.), but if not, let a e, b e, c e, d E Fig. 130. 
be four harmonicals, and let them meet any ftraight 
line a d in a, b, c, d. Through b draw the ftraight 
line g h parallel to d d, and let it meet the ftraight 
line e a in g, and e c in H. Then, by the preceding 

Lemma, 


158 


LEMMAS FOR THE FOURTH BOOK 

Lemma, g h is bife&ed in B ; and (4. vi.) d a 
d e : b G or b h ; and de:bh::dc;cb. 
quently (11. v^da:ab;:dc:cb, 





: A B : : 
Confe- 



* 

. 'J 


A GEO- 


A 


GEOMETRICAL TREATISE 

OF 

CONIC SECTIONS. 


BOOK IV. 

Of Jimilar Sections, general Properties, Circles having the 
fame curvature with the SeElions in given points, and 
ofjlraight lines cut harmonically hy the SeElions. This 
Book alfo contains Problems ifeful in the Theory of 
AJlronomy, and Methods of finding two mean propor¬ 
tionals and of trifeding an angle , by means of the Sec¬ 
tions. 


DEFINITIONS. 

I. 


TL\vO fegments of conic fe&ions are called Similar 
Segments, if a rectilineal figure can be infcribed in one 
of them fimilar to any redilineal figure infcribed in 
the other. 


II. 


Two conic fe&ions are called Similar Sedions, if a 
redilineal figure can be infcribed in one of them fimi- 

lar 







l6o 


BOOK 

IV. 


Fig. 117. 
118. 


OF SIMILAR SECTIONS. 

Iar to any reCtilineal figure infcribed in the other : and 
two conic feCtions are alfo called Similar if a fegment 
can be taken of one of them fimilar to a fegment of the 
other. 

III. 

If a ftraight line touch two conic feCtions in the fame 
point, the two feCtions are laid to touch one another in 
the fame point. 

IV. 

If a circle fo touch a conic feCtion in any point, that 
no other circle, touching it in the fame point, can pafs 
between it and the feCtion, on either fide of the point 
of contaCt, it is laid to have the fame Curvature with the 
Section in the point of contact) or it is faid to be the OJ'cu- 
lating Circle for that point . 

PROP. I. 

Any two parabolas are fimilar to one another) and the fi¬ 
milar reElilineal figures infcribed in them are to one ano¬ 
ther as the fquares of the parameters of the axes . 

Let abc, a b c be two parabolas, of which be, be 
are the axes, and p, p their parameters; and let abc i> 
be any reCtilineal figure infcribed in the one parabola ; 
a reCtilineal figure fimilar to abcd may be infcribed 
in the other, and the fimilar reCtilineal figures infcribed 
in them are to one another as p 2 to p z . 

For draw from b } the vertex, the ftraight line b a to 
the curve, fo that the angle eh a may be equal to the 
angle eba. Draw be to the curve, fo that the angle 
e b c may be equal to the angle ebc. Draw B d, and 
draw b d to the curve, fo that the angle e b d may be 
equal to the angle e b d ; and draw a d. Let a e be 
an ordinate to the axis b e, and let a e be an ordinate 
to the axis b e % Then, as the angles at e and e are 

right 




OP SIMILAR SECTIONS. l6l 

right angles, the triangles a b e, a b e are equiangular, BOOK 
and (4. vi.) be : ea : But, by the fixth 1V * 

Definition Book III. be:ea::ea:p; and b e :' 
e a : : e a : p. We have therefore the two following 
ranks of magnitudes, in which the magnitudes taken 
two and two in the fame order have the fame ratio to 
one another $ 

BE:e a : p 

be : e a : p 

and confequently (22. v.) b e : p :: be:p; or by al¬ 
ternation B e : b e : : p : p. But (4. vi. and altern.) 

B E : b e : : b A : b a, and therefore (11. v.) ba:^:: 
p : p. In the fame way it may be proved, that b c : 
b c : : p : p, and that b d : b d : : p : p; and therefore 
(n.v.) bc : be: :bd : b d, and by alternation b c : 

B d : : b c : b d. But as the angles ebd, eb d are 
equal, and the angle ebc equal to the angle eb c, the 
angles dbc, d b c are equal. Confequently (6. vi.) 
the triangles c b d, cbd are equiangular, and (4. vi. 
and altern.) CD:c^::Bc:^,or p to f In the 
fame way it may be proved that a d : a d : : p : p- 3 and 
therefore the rectilineal figure abed is fimilar to the 
rectilineal figure abcd. The parabolas abc, a bc 
are therefore fimilar according to the firft and fecond 
Definitions, and it is evident (Cor. 2. 20. vi.) that the 
rectilineal figures abcd, abed are to one another as 
the fquares of their homologous fides, or (11. v.) as p r ' 
to p\ 

Cor. 1. It is evident from the above that fimilar rec¬ 
tilineal figures may be inferibed in the fimilar parabolic 
fegments abcd, abed , of which the homologous 
fides will be to one another as p to />, and which will 
be deficient from the parabolic fegments by fpaces lefs 
than any given. The fimilar parabolic fegments them- 
felves will therefore be to one another as p* to />*. 

M 


Cor. 



OF SIMILAR SECTIONS^ 


i6z 

BooR Cor . 2. In two parabolas the parameters of dian^ 

1V> ters, which contain equal angles with their ordinates,, 
are to one another as the parameters of the axes. For, 
the reft remaining as above, let A g, touching the pa¬ 
rabola A b c in A, meet the axis b e in g ; and let ag , 
tduching the parabola a be in a , meet the axis be in g. 
Let f be the focus of abc, and f the focus of a b c, 
and draw af, af Then, as bf is one fourth of p, 
and bf one fourth of />, by the above, (and 15. v.) ab: 
bp :: a b : bf; and therefore (6. vi.) the triangles 
A B f, a bf are equiangular, and af:bf : : af: bf; 
or, by Cor. 2. Prop. XI. Book III. (and 15. v.) the pa¬ 
rameter of the diameter palling through a is to p as the 
parameter of the diameter palling through a to p. 
Again, by Cor. Prop. IX. Book III. the triangles af g, 
afg are ifofceles, and, as above, the angles a f g, af g 
are equal. The angles a g f, agfaxo, therefore equal $ 
and, as the diameters of a parabola are parallel, the an¬ 
gle a g f is equal to the angle which the diameter 
palling through a contains with its ordinates, and the 
angle agf equal to the angle which the diameter palf- 
ing thrqugh a contains with its ordinates, bv Prop. IL 
Book III. 


PROP, II. 

Two ellipfes , or two hyperbolas , are fimilar to one ano 
tber i f two conjugate diameters in the one be propor * 
tional to two conjugate diameters in the other , and the 
firft two and the other two contain equal angles . On 
the contrary , if two ellipfes , or two hyperbolas , be jimi± 
lar to one another , two conjugate diameters in the one 
will be proportional to two conjugate diameters in the 
other , provided the frjl two and the other two contain 
equal angles 4 


Let 



£)P SIMILAR SECTIONS* 

Let b h, b h be two ellipfes, or two hyperbolas, and 
in the one let a b a diameter be to d e its conjugate 
as, in the other, a b a diameter is to de its conjugate, 
and, c, c being the centers, let the angle Dcebe equal 
to the angle dcb \ then the ellipfe b h is fimilar to the 
ellipfe b h , and the hyperbola b h is fimilar to the hy¬ 
perbola b b. On the contrary, if the ellipfe or hyper¬ 
bola b h be fimilar to the ellipfe or hyperbola b h , and 
if the angle dcb contained by ab, de, two conjugate 
diameters in the one, be equal to the angle dcb con¬ 
tained by ab, de, two conjugate diameters in the other; 
then A b is to d e as a b to de . 

Part I. Let b h m l k be any rectilineal figure in- 
feribed in the ellipfe or hyperbola b h, and from c the 
center draw the ftraight lines c h, cm, cl, c k. 
Again, at c the center of the ellipfe or hyperbola b b, 
make the angles b c h, he m, m cl, l eh, equal to the 
angles bch,hcm, m c l, l c k, each to each ; and 
let the points h , m, l, k be in the curve of the feCtion. 
Then the ftraight lines b h, hm , m /, Ik, kb being 
drawn, the reCtilineal figure bhmlk inferibed in the 
ellipfe b b is fimilar to b h m l k inferibed in the el¬ 
lipfe b h ; and the rectilineal figure bhmlk inferibed 
in the hyperbola b b is fimilar to bhmlk inferibed 
in the hyperbola b h. For draw hg an ordinate to 
a b, and hg an ordinate to ab. Then, as the angles 
d c G, deg are equal, and as, by Cor. 2. Prop. IV. 
Book II. h G is parallel to d c, and bg parallel to d c , 
the angles (29. 1.) at g and g are equal. The angles 
H c g , beg are alfo equal, by conftruCtion, and there¬ 
fore the triangles 11 c G, h eg are equiangular. Con- 
fequently (4. vi.) G H : C G : : g h : c g; and G H 2 : 
cg 2 : : gb z : eg 2 , and (16. v.) gh 2 : gb 2 : : c g 2 : 
eg 2 . But, by hypothefis, (and 15. v.) cb : CD : : 
c b : c d , and therefore cb 2 :cd z : : c b 1 : c d 2 ; and, 
m 2 by 


BOOK 

IV. 


Fig. 119- 
120 . 
I2T* 
122 . 



OF SIMILAR SECTIONS, 


164, 

BOOK 

IV. 


by Prop. V. Book It cb 1 ; cd j : : ag x gb r gh^ 
and c b z : c d z : : ag X g b : g b\ Hence (u. v.) a g 
X G B : g H a : : ag X gb : g b* ; and (16. v.) a g x 
G b : ag X g b : : g h z : gh z . Confequently (11. v.) 
AGXGB: a g x g b : :CG 2 : c g z } and therefore 
(16. v.) A G X G B ^ c g 2 : : ag X g b : cj* 2 , and (by 
5. and 6. ii. and 17. and 18. v.) c b 2 : c g 2 : : c b 2 z 
cg z . We have therefore (23. vi.) c b : c g : : c b : eg; 
and (16. v.) cb:c^:;cg: eg. Again, by the fi- 
milar triangles, c g : c g : : c u : c b •, and therefore 
(11. v.) c b : c b : : c h : cb, and by alternation c b : 
c h : : c b : c b. Confequently (6. vi.) the triangles 
b c h, bob are fimilar. In the fame way it may be 
proved that c B : cb \ : c m z cm; and therefore (ii.v.) 
that c h : cb : : c M : c m. Confequently, by alter¬ 
nation, (and 6. vi.) the triangles mch >mcb are fimi¬ 
lar ; and in the fame way it may be proved, that the 
triangle l cm is fimilar to the triangle l c m, the trian¬ 
gle lek to the triangle lck, and the triangle l cb to 
the triangle kcb. The redilineal figures b h m l k 
lb mlk (20. vi.) are therefore fimilar, and the fe&ions 
B h, b h are therefore fimilar, according to the firfl: and 
fecond Definitions. 

Part II. If a b be not to d e as A b to d e, let a B be 
to D e as a b to a ftraight line greater or lefs than de ; 
and with this firaight line as a conjugate diameter, 
and A b as a tranfverfe diameter, fuppofe an ellipfe or 
hyperbola to be defcribed. Then this ellipfe or hyper¬ 
bola will fall without or within the fe&ion b h of the 
fame name, and, by the preceding part, a redilineal 
figure may be inferibed in the fe&ion, at prelent fup- 
pofed to be defcribed, fimilar to the redilineal figure 
B h m l k. But as, by hypothefis, the fedions b h, 
l b are fimilar, a redilineal figure may be inferibed in 
the fedion bb fimilar to the redilineal figure bhml k, 

ac- 



OF SIMILAR SECTIONS. 


26$ 

according to the firft and fecond Definitions, Let this BOOK 
infcribed figure be bb mlk. Then (21. vi.) in the fee- IV ‘ 
tion, having a b for its tranfverfe diameter, and falling 
either without or within the fe&ion b b, a re&ilineal 
figure may be infcribed fimilar to bhmlk ; which 
(from Def. i. vi. and 21. i.) is evidently abfurd. Con- 
fequently, the fe&ions b h , b b being fimilar, and the 
angles dcb, deb equal, A b is to D e as a b to d e. 

Cor. 1. In fimilar ellipfes, or fimilar hyperbolas, di¬ 
ameters which contain equal angles with the axes are 
to one another as the axes. For if a b, a b be the 
tranfverfe, and d e, d e the conjugate axes, then the 
angles b c h , b e b being equal, c b is to c H as c b to 
c b, as was above demon ftrated. 

Cor. 2. From the above (and Cor. 2.. 20. vi.) it is 
evident that fimilar rectilineal figures may be infcribed 
in fimilar ellipfes, or in fimilar hyperbolic fegments, 
which (hall be to one another as the fquares of the 
tranfverfe, or as the fquares of the conjugate axes. 

Cor . 3. Similar ellipfes, and fimilar hyperbolic feg- 
ments, are to one another as the fquares of their tranf¬ 
verfe, or as the fquares of their conjugate axes. This 
is evident from the preceding Cor. (and 2. xii.) as a 
rectilineal figure may be infcribed in an ellipfe, or in a 
hyperbolic fegment, which {hall be deficient from the 
ellipfe, or hyperbolic fegment, by a fpace lefs than any 
given fpace. 

Cor. 4. The angles contained by the afymptotes of 
fimilar hyperbolas are equal to one another; and if the 
angles contained by the afymptotes of two hyperbolas 
be equal, the hyperbolas will be fimilar. For let a b, 
d e be the axes, c the center, and c s an afymptote of 
the hyperbola b h ; and let a b, d e be the axes, c the 
center, and c;an afymptote of the hyperbola b h. Let 
b s touch the hyperbola b h in the vertex b, and meet 
M 3 the 



166 


OF SIMILAR SECTIONS. 


BOOK the afymptote in s ; and let b s touch the hyperbola 
bb in the vertex b, and meet the afymptote in s. Then 
the angle CBS is equal to the angle c b s, as each is a 
right one, and, by Cor. 3. Prop. XV. Book III. bs is 
equal to ce, and bs is equal to ce; and therefore, if the 
hyperbolas be fimilar, by the fecond part of this Prop, 
(and 15. v.Jcb'.bs::^:^, Confequently (6. vi.) 
the angle b c s is equal to the angle b cs. On the con¬ 
trary, if the angle bc s be equal to the angle bcs , 
then (4. vi.) cb:bs : : cb : b s ; and therefore, by 
the firft part of this Prop, the hyperbolas BU y bb are 
iimilar, 

PROP. III. 

If two ellipfes , or two hyperbolas , be fimilar , the tranf¬ 
verfe axis in the one will be to the di/lance between the 
foci as the tranfverfe axis in the other to the difiance 
between the foci . On the contrary , two ellipfes , or two 
hyperbolas , will be fimilar , if the tranfverfe axis in the 
one be to the difiance between the foci as the tranfverfe 
axis in the other to the difiance between thefoci . 

Fig. 119. Part I. Let b h, bh be two fimilar ellipfes^ and b h, 
b h be two fimilar hyperbolas. Let A b, d e be the 
*2*. tranfverfe and conjugate axes, c the center, and f, o 
the foci of the one ; and let a b, d e be the tranfverfe 
and conjugate axes, c the center, and f 0 the foci of 
the other ; then a b is to f o as a b is to fo. 

For in the ellipfes draw e o, eo, but in the hyper¬ 
bolas draw e a, ea . Then, by Cor. 3. and Cor. 3. 
Def. XI. Book II. in the e ipfes e o is equal to c a. 
and eo is equal to ca; but in the hyperbolas e a is 
equal to c o, and e a is equal to c 0. Hence, as the 
fecSHon b h is fimilar to the fe&ion b h, by the fecond 
part of Prop. II. in the ellipfes eo : ec : : eo : ec; 

but 



OF SIMILAR SECTIONS, 

but in the hyperbolas c a : c e :: c a : c e. Confer 
quently, as the angles at c and c are equal, being right 
angles, the triangles (7. vi.) e c o, eco in the ellipfes 
are equiangular; and in the hyperbolas the triangles 
(6. vi.) ace, ace are equiangular. Hence in the el¬ 
lipfes e o, or its equal e a, is to c o as eo, or its equal 
c a , is to co ; but in the hyperbolas c a is to a e, or 
its equal c o, as ca is to a e, or its equal co. 

Part II. On the contrary, the reft remaining as above* 
if it be allowed, either in the ellipfes or hyperbolas, 
that c a is to c o as c a to c 0, then it may be proved, 
as above, (by 7. vi.) that in the ellipfes the triangles 
E c o, e c 0 are equiangular, but in the hyperbolas that 
the triangles ace , ac e are equiangular. Confequent- 
3 y in the ellipfes e o, or its equal c a, is to c e as e 0, 
or its equal c a, to c e ; and in the hyperbolas c A is to 
c e as c a to c e. Hence this part of the Cor. is evident 
from the fir ft part of Prop. II. 

Cor. If b h, b h be two ellipfes, or two hyperbolas, 
of which the foci are o, f in the one, and o,f in the 
other, and if the triangles m of, mof be equiangular; 
then if the fe&ions be fimilar, and the point m be in 
the curve of b h, the point m will be in the curve of 
b h. For (4. vi.) om:mf : : om : mf; and therefore 
(18. v.) in the ellipfes om+mf:mf::o?» + mfz 
mf; and (17. v.) in the hyperbolas o m — m f : m f : : 
om — vif : inf. Again, (4. vi.) either in the ellipfes or 
hyperbolas, m f : f o : : mf :fo. Confequently, in 
the ellipfes, 

om + mf:mf:fo 

0 m + mf ‘ mf : f 0; 

and therefore (22. v.) om + mf : f o : zom+mf : fo a 
Alfo, in the hyperbolas, 

om — mf:mf:fo 
cm — mf: mf :f 0 ; 

M 4 and 


i6y 

BOOK 

IV. 



x68 


OF SIMILAR SECTIONS* 


BOOK 

IV. 


and therefore (22. v.) o m — m f : f o : : om — mfz 
fo. But, the reft remaining as in the Proposition, by 
Prop. XIII. Book II. om + mf in the ellipfe, but 
o m — m f in the hyperbola, is equal to a b, and as 
the fe&ions are fimilar, by this Prop, a b : F o : : a b : 
fo. Confequently (11. v.) in the ellipfe ab :fo : : 
om-\-mf : fo ; and in the hyperbola a b :fo : : 0 m — 
m f :fo . In the ellipfe, therefore, (14. v.) 0 m f is 
equal to a b , and in the hyperbola om — mf is equal to 
a b. Confequently the point m is in the curve of the 
fe&ion b b 3 by Cor. 1. Prop. XIV. Book II. 

SCHOLIUM. 

Having explained methods for defcribing conic fec- 
tions, and demonftrated the principal properties of fi¬ 
milar fe£tions, it may be proper, in this place, to give 
an explanation of fome palfages in the fourth fe£tion of 
the firft Book of the Principia. The reader, who 
thinks luch explanations neceflary, is fuppofed to have 
that juftly celebrated work before him whilft he perufes 
this Scholium, as he is here referred to the figures 
which it contains. In the following explanations the 
Lemmas and Propofitions of the above-mentioned Sec¬ 
tion of the Principia are printed in capital letters, to 
diftinguilh them from fuch parts of this treatife as are 
referred to. 

LEMMA XV. This is fully explained in Cor. 2. 
Prop. XV. Book II. 

PROPOSITION XVIII. This is evident from Prop. 
XIII. and Cor. 2. Prop. XV. Book II. 

PROPOSITION XIX. The reafons for the defcrip- 
tions*here delivered are eafily deduced from Cor. 3. 
Prop. X. and Cor. 3. Prop. VIII. Book III. and Lem¬ 
ma IX. and it is evident that f i will be the dire&rix 
of the parabola. 


PROPO- 



OF SIMILAR SECTIONS. 

PROPOSITION XX. An ellipfe or hyperbola is 
faid to be given in fpecies when it is fimilar to a given 
ellipfe or hyperbola, or when the ratio of the axes to 
one another is given. When the ratio of the axes to 
one another is given, the ratio of the principal or tranf- 
verfe axis to the diftance of the foci is alfo given, or is 
conftant, according to Prop. III. 

The remaining part of Cafe i. PROP. XX. is evi¬ 
dent from Cor. i. Prop. VIII. Book III. and Lemma 
IX. 

Cafe 2. That a ftraight line bife&ing v v at right 
angles will pafs through the other focus, of the ellipfe 
or hyperbola, has been proved in Cor. 3. Prop. XV. 
Book II. and that the circumference of a circle de- 
fcribed upon k k as a diameter will pafs through the 
fame focus has been proved in the 4th Cor. to the 
fame Proportion. The whole of Cafe 2. is therefore 
evident from the Corollaries already mentioned, and 
from Prop. XIIJ. and XV. Book II. 

Cafe 3. That a circle defcribed upon k k as a diame¬ 
ter will pafs through the other focus is evident from 
Cor. 4. Prop. XV. Book II. and that v r will pafs 
through the fame focus is evident from Prop. XV* 
Book II. 

Cafe 4. That v h is equal to a l is evident from Sir 
Ifaac Newton’s demonfration. Recurring to propor¬ 
tions previoufly hated s 1-1 : sh : : s p : s p, and there¬ 
fore sh : s p : : sh : sp \ and as the angles p s H,/> sh 
are equal, the triangles p s h, psb are fimilar. Again 
s v : sp : : s b : sq ; and on account of the fimilar tri¬ 
angles v s p, h s q i s v : s p : : bs : sq. Confequently 
(11. v.) s y : s p : : sv : sp , and by inverfion s p : s v : : 
s p : sv. From the above therefore 
s H : s p : s v 
s b ; s p 1 sv, 

and 


169 

BOOK. 

IV. 



170 


OF SIMILAR SECTIONS'. 


BOOK. 

IV. 


and (22. v.) s h : s v : r sh : s v, and as the angles 
v s H, v s h are equal, the triangles (6. vi.) v s h, v s h 
are fimilar. From hence, and Prop. III. and its Cor, 
the proceedings in this cafe are evidently juft. 

LEMMA XVI. Cafe 1. By Cor. 1. Prop. VIII. Book 
III. p r is the dire&rix of the hyperbola, of which n m 
is the tranfverfe axis, and a, b the foci, and therefore 
A p is given; and by the fame Cor. z R i A z : : M n : 
A b. Let d denote the difference between az , cz, and 
then d by hypothefis is given. For the fame reafons 
a a is given, s q being the dire&rix to the other hy¬ 
perbola 5 and, as above, a z : z s : : a c : d. Confe- 
quently, by the fifth Lemma, (and 1. vi.) z R : z s : : 
M n x A c : A B X D and as m n, a c, a b, and i> 
are given, the ratio of z r to z s is given. Again, 
Radius : fine < zts r: tz :'zs; and therefore by 
the above, and the fifth Lemma, (and 1. vi.) Radius x 
D : fine <zts X a c : : t z : a z. But T z and t s 
being given in pofition, the fine of the angle zt s is 
given, and therefore the ratio of t z to a z is given. 
The ftraight line t a is given, and alfo the angle atz, 
and a z : t z : : fine <atz : fine <taz; and there¬ 
fore the angle t a z is given. Confequeotly the tri¬ 
angle a t z is given. By introducing fome additional' 
ftraight lines into the figure, the triangle might be af- 
eertained geometrically. 

The other two cafes need no explanation; nor do 
the remaining parts of the Section, as the feveral cafes 
in PROP. XXL may be folved by the preceding LEM¬ 
MA. The SCHOLIUM is evident from Prop. VIII. 
Book III. 


PROP. IV. 

If two ftraight Vines touching a conic fettlon, or oppofite 
hyperbolas , meet a ftraight line which cuts the feViion, 

appo - 



OP GENERAL PROPERTIES. 


171 


oppoftte hyperbola, or oppojite hyperbolas, and is parallel BOOK 
to the Jlraight line joining the points of contact, the feg - 1V ‘ 

ments of the fecant between the curve or curves and the 
tangents will be equal to one another: and if two 
Jlraight lines touching a conic feStion meet a ftraight line 
which touches the feElion, or oppofte hyperbola, and is 
parallel to the ftraight line joining the points of contact, 
the fegments of the laft mentioned tangent between the 
point of contact and the tangents which it meets will be 
equal to one another . 


If the two tangents be parallel, this Prop. Is the 
fame as the Cor. to Prop. I. Book II. and this cafe has 
been already demonftrated. Let the two ftraight lines 
therefore a e, d e touch the conic fe&ion a d, or the 
oppofite hyperbolas a, d, in the points A, d, and meet 
one another in e, and, firft, let them meet in the points 
h, k the ftraight line h k, which is parallel to A d, 
and cuts the curve of the fe&ion, the oppofite hyper¬ 
bola, or the curves of the oppofite hyperbolas, in the 
points f, g ; the fegments h f, k g are equal to one 
another. 

For let the ftraight line e m bife& ad in m, and 
meet the ftraight line h k in n ; and then, by Cor. 
I. to Prop. VI. Book III. e m is a diameter, and 
therefore, as ad,fg are parallel, f g is biletted in n, 
by Cor. i. to Prop. II. Book III. But (29. i. and 4. 
vi.) e M : E N : : A M ; H N, and e m : E n : : d m : 
k n. Confequently (11. v.) A M : H N : : D M : K N, 
and therefore (14. v.) h n is equal to k n, and h f 
equal to kg. 

The reft remaining as above, let the tangents a e, 
d e now meet the ftraight line l 1 in the points l and 
1, and let l i be parallel to a d, and touch the fe&ion, 

or 


Fig. 131. 
13*. 

133 * 

334 * 


Fig. 131 
132- 
* 34 “ 



OF GENERAL PROPERTIES. 


* 7 * 

BOOK or oppolite hyperbola, in the point b ; the fegments 
IV ' lb, bi are equal. 

For as u, ad are parallel, by Prop. II. Book III. 
a diameter palling through b will bife& a d, and, by 
Prop. VI. Book III. it will pafs through e. The dia¬ 
meter e M therefore palfes through b the point of con¬ 
tact, and (4. vi.) em: eb u am : lb. Alfo e m : 
B b : : D M : 1 B, and therefore (n. v.) A M : l b : : 
DM : IB. Confequently (14. v.) L B is equal to 1 b. 

Pig. 131. Cor. 1. If two parallel ftraight lines as a d, g f cut 
a conic fe&ion, or oppolite hyperbolas, in a, d and g, 
134- f, and ftraight lines a g, d f joining their extremities 
meet in o, ft the ftraight line o a, which cuts the fee- 
tion, or oppolite hyperbolas, in p, r, and is parallel to 
ad, G f ; the fegments o r, a p are equal. For let 
the tangents a e, d e meet o a in t and s ; and then 
(4. vi.) A h : A t : : h G : t o, and d k : D s : : k f ; 
s ft. But, on account of the parallels, a h : A t : : 
D k : d s, and therefore hg:to::kf:sq; and 
as, by this Prop, h g is equal to k f, and t r to s p, 
it is evident (14. v.) that o r is equal to a p. 

Frg. 131. Cor . 2. The reft remaining as above, let A G meet 

the tangent l 1 in v, and let d f meet it in y, and the 
fegments v r, y b will be equal. For it may be de- 
monllrated, as in the preceding Cor. that l v, i y are 
equal; and confequently, as lb, ib are equal, the 
legment v b is equal to the l’egment y b. 

PROP. V. 

If a trapezium he inferibed in a conk fehlion, or oppofit? 
hyperbolas , and its Jides be indefinitely produced , and if 
from any point in the curve two firaight lines be drawn 
parallel to two adjacent Jides of the trapezium arid meet 
the obpnjiteJides ; the re^ angles under theJegments of thej'e 

firaight 



OP GENERAL PROPERTIES. 


173 


jlraight lines , between the point in the curve and the op¬ 
pofite Jides of the trapezium, will be to one another as 
the fquares of the femidiameters parallel to them, or the 
fquares of the tangents parallel to them, and meeting 
one another , 

Let A b D c be a trapezium infcribed in a conic fec- 
tion, or oppolite hyperbolas, and let its fides be indefi¬ 
nitely produced, and from any point e in the curve let 
the ftraight lines e n, e h be drawn parallel to the ad¬ 
jacent fides a b, ac, and let e n meet the oppofite 
fides A c, b d in a and n, and e h meet the oppofite 
fides a b, c d in r and h ; the rectangles a e n, h e r 
are to one another as the fquares of the femidiameters 
parallel to e n, e h, or the fquares of the tangents pa¬ 
rallel to them and meeting one another. 

For if the oppofite fides a c, b d be not parallel, let 
the ftraight lines B d, df be drawn parallel to a c or 
H R ; and let b d meet the curve again in d, and let it 
meet e n in n. Let d f meet the curve again in f, 
and the ftraight line a b in g. Draw the ftraight line 
c d, and let it meet the ftraight line H R in h, and the 
ftraight line d f in s. Let the ftraight line h r meet 
the curve again in t. Then, by Cor. 1. Prop. IV. s d, 
f G are equal, and h t is equal to e r, and confequent- 
3 y h e, t R are equal. The rectangle h e r is there¬ 
fore equal to the rectangle t r e ; and, on account of 
the parallelograms be, a e, the re&angle ft e n is 
equal to the redangle arb. The rectangles T r e, 
arb are therefore to one another as the rectangles 
h e r, a e n ; and confequently, by Prop. V. Book. II. 
and Prop. XIII. Book I. thefe rectangles are to one 
another as the fquares of the femidiameters parallel to 
11 r, e N, or as the fquares of tangents parallel to h r, 
fc n, and meeting one another. Again, on account of 


BOOK 

IV. 


Fi£. 123. 
124. 



m 


OF GENERAL PROPERTIES, 


book the equiangular triangles h c £, dcs, h £ : s d : S 
* c h : c s ; and therefore, on account of the equals 
and parallels, Hi>:FG::Eft:AG. Alfo, on account 
of the equiangular triangles n b «, d b g, n b or (34. i.) 
£r:dg::ntz:gb; and therefore, by the fifth 
Lemma, hZ>xer:dg xgf: : eq. X n ; a g x 
g E \ and by alternation h£xer:eq.XN«:: 
dgxgf:agxgb. But, by Prop. V. Book II. 
and Prop. XIII. Book I. D g x g f and a g X g b are 
to one another as the fquares of the femidiameters pa¬ 
rallel to h R, E n, or as the fquares of tangents paral¬ 
lel to H r, e n, and meeting one another. By the 
above therefore (and 11. v.) £ e x e r : q e X e n: : 
H/>XER:EaXNw; and by alternation h e x e r : 
i^XEr::gexe«:qe X n w ; and therefore 
(1. vi.) b e : H h : : E n : N n, and (17. and 18. v.) 
H e : h h : : e n : n n. Confequently by alternation 
(and 1. vi.) hex er :en x eg :: h/;x er: n« 
X e a ; and therefore, by the above, (and 11. v.) he 
X e r and e n x e a are to one another as the fquares 
of the femidiameters parallel to h r, e n, or as the 
fquares of tangents parallel to h r, e n, and meeting 
one another. 

Fig. 125. If the ftraight lines a b, c d cut one another like 
the diagonals of a trapezium, as in Fig. 125. and the 
reft be as exprefled above in the particular enunciation; 
then it may be proved, as above, that o, e X e n and 
h e x e r are to one another as the fquares of the fe- 
midiamers parallel to e n, e h, or the fquares of the 
tangents parallel to them, and meeting one another. 

It is evident that the method of demonftration and 
conclufion will be the lame, if one of the ftraight lines 
h r, b d, d f, or even two of them, be tangents. 

Fig. 123. Cor . 1. If the points A, b, c, d remain fixed, and 

125. & e point e with the ftraight lines en,e h, always pa¬ 
rallel 



OF GEN EH A L PROPERTIES* 


rallel to a b, a c, be carried round the fe&ion, or op- boor 
polite hyperbola, in every fituation of e the rectangles IV * 
gen, her will be to one another in the fame ratio. 

Cor. 2. The points a, b, c, e remaining fixed, if the 
point d, the interfection of the firaight lines bd, CD, 
be carried round the feCtion, or oppofite hyperbola, in 
every fituation of d the fegments e h, e n will be to 
one another in the fame ratio. For the rectangles 
h e r, q e n in every fituation of d will be to one 
another in the fame ratio, and therefore as e r, q e re¬ 
main fixed, the Cor. is (i. vi.) evident. 

Cor. 3 . The reft remaining as above, draw b c, and Fig- 125- 
let it meet e h in l. In the ftraight line e q, and on 
the fame fide of e h with the point n, let the point t 
be taken, and let h e be to e n as le to et; and 
then b t being drawn, it will touch the feClion. For, 
if it be poflible, let it meet the curve again in v ; and 
v c being drawn, let it meet eh in x. Then, by Cor. 

2 . x E ! e t : : he:en, and therefore (11. v.) le: 
e t : : x e : e t, and l e, x e are equal: which is ah- 
furd. The ftraight line b t therefore touches the feciion. 

Cor . 4 . Hence, the reft remaining, if the ftraight 
line b t touch the feCtion in b, and meet the ftraight 
line e n in t, h e will be to e n as l e to e t. 

Cor. 5. If the ftraight lines ar, c ii touch the fee- Fig. 12?- 
tion in a, c, and from e, a point in the curve, e a be 
drawn parallel to ar, and meet a c, the line joining 
the points of contaCl, in q, and h e r be drawn paral¬ 
lel to A c, and meet the tangents in r and h ; then 
the reClangle her and the lquare of eg will be to 
one another as the fquares of the femidiameters paral¬ 
lel to h r, q e, or as the fquares of the tangents pa¬ 
rallel to h r, a E, meeting one another. For if h r 
meet the curve again in t, then, by Prop. IV. h t, 
e r are equal, and the rcClangles h e r, t r e are 

equal: 



1J6 


BOOK 

IV. 


Fig. 135- 
136. 


of general Properties# 

equal: and (34. i.) as E a is equal to A r, the Cor. is 
evident from Prop. V. Book II. and Prop, XIII# 
Book I. 


PROP. VI. 

If a trapezium he infcribed in a conic fe&ion, or oppofte 
hyperbolas , and from any point in the curve aft raight 
line be drawn in a given angle to each of the Jides , or 
the fdes produced , the rectangle under the lines drawn 
to two oppoftefdes will be to the rectangle under the 
lines drawn to the two other oppofte fdes in a given- 
ratio. 

Let A b d c be a trapezium inferibed in the conic 
fe£tion e A b d Cj or in the oppolite hyperbolas a d c, 
e b, and from any point e in the curve let the ftraight 
lines el,em,eo,ek be drawn in given angles to 
the fides a b, c d, b d, a c, each to each ; the rectan¬ 
gle under e l, e m, drawn to the oppofite fides a b, 
c d, is to the rectangle under eo,ek, drawn to the 
other two oppofite fides b d, a c, in a given ratio. 

For from any other point e in the curve of the fec- 
tion, or in the curve of either of the oppofite hyperbo¬ 
las, let the ftraight lines el> e m, e o, e k be drawn pa¬ 
rallel to e L, e m, e o, e K, each to each, and to the 
fame tide of the trapezium each to each. Through 
the point e let the flraight lines ax, hr be drawn 
parallel to the adjacent fides A b, a c, and meeting the 
fides a c, 11 d, a b, d c, in the points a, n and r, h. 
Through the point e draw the ftraight lines qn , hr pa¬ 
rallel to the ftraight lines a n, h r, or to the fides a b, 
a c, and meeting the fides ac, bd,ar,dc in the 
points q , n and r, h, Then (29. i. and 4. vi.) e r : 
e r : : E L : e l ; and e h : e h : : E M : e m. Confe- 
quentlv, by the fifth Lemma, hr xeh : er X eh : : 

E L 



OF GENERAL PROPERTIES. 


x /7 

EL X em \ el x e m . Again (29. i. and 4. vi.) e q : BOOK 
eq : : E k : e k ; and en:£«::eo:£©> and there- IV ’ 
fore, as before, Eax en:££X*?z::ekxeo: 
ek x eo. But, by Cor. 1. Prop. V. erxeh:£?*x 
e h : : n ql x ix : e q x e 71', and therefore (11. v.) 
elxem : e l x e m : : E k x E o : £ £ x £ 0. Con¬ 
sequently, by alternation, the reCtangle under e l, e m 
is to the re&angle under e k, e o in the conftant or 
given ratio of the reClangle under el, em to the rect¬ 
angle under eh, e 0. 

Cor. If two ftraight lines ah, CH touch a conic Fig* i* 7 » 
feCtion in a, c, and if from any point e in the curve 
ftraight lines e l, e m, e k be drawn in given angles 
to the tangents, and a c joining the points of contaCI, 
the reClangle under e l, e m and the fquare of ek 
will be to one another in a conftant or given ratio. 

For take any other point e in the curve, and through 
e, e draw h r, h r parallel to a c, and let them meet 
the tangents in h, r and b , r; and draw E a, e q pa¬ 
rallel to a r, and let them meet a c in q, q . Then 
by fimilar triangles, as above, erxeh : e r x e b : : 
el x em : e l x e m; and by Cor. 5. Prop. V. e r x 
e h : c r x e h : : e a 2 : eq^. But on account of the 
parallel lines, the triangles e k a, ekq are ftmilar, and 
e a 2 : eq z : : e k 2 : ek 1 . Consequently (11. v.) e l x 
em : cl x em : \ ek 2 : e k 2 . 

PROP. VII. 

The curve of a conic fefiion cannot meet the curve of ano¬ 
ther conic fektion , or the curves of oppofte hyperbolas, in 
more than four points . 

For, if it be poflible, let the curve of a conic fe&ion Fig> 137> 
meet the curve of another conic feCtion, or the curves 
of oppoftte hyperbolas, in the points a, b, d, c, e ; 

n and 



178 


OF GENERAL PROPERTIES. 


book and draw ab, b d, d c, c a. Let the draight lines 
R * e n, e h be drawn parallel to a b, a c, and let them 
meet bd, d c in n and h. Let the draight line b f 
be drawn, meeting the curves again in f, i, and the 
draight line e n in n. Let the draight lines f c, i c 
be drawn, and let them meet the draight line e h in 
k and l. Tlien, by Cor. 2. Prop. V. H e : e n : : 
l e : e n, and h e e n : : k e : E n. Confequently 
(11. v.) l e : E n : : k E : E ?i, and (14. v.) l e, k e 
are equal: which is abfurd. The curve of the conic 
fedion, therefore, does not meet the curve of the 
other, or the curve of oppofite hyperbolas, in five 
points. 

Lg. 138. Cor. 1. If two conic fedions touch one another, they 
will not meet each other in three other points. For, 
if it be podible, let the two fedions have the common 
tangent in the point b, and meet one another in a, e, 
c, and of thefe let e be the intermediate point. Let 
B A, b c, c A be drawn; and let e t, e h be drawn 
parallel to ba,ac, and let them meet b t, bc in t 
and h. Through the point of con tad b let b d be 
drawn, meeting the curves in d and d } and the draight 
line e t in n ; and let d c, d c be drawn, meeting e h 
in 1 and l. Then, by Cor. 4. Prop. V. h e is to e t 
as l e to e n, and 1 e is to e n in the fame proportion, 
and therefore (9. v.) ie, le are equal: which is ab¬ 
furd. The two points d, d therefore coincide, and the 
two fedions meet in the five points A, b, d, c, e ; 
which by this Prop, is impodible. 

Fig. 139. Cor. 2. two conic fedions a f d, xfv> touch one 
another in the points A, d, they will not meet one 
another in any other point. For, if it be poffible, let 
them meet one another in the point 1, and let the 
draight line \ f be drawn, meeting the tangents a b, 
d g in b, g, and the curve of the fedion a f d in f. 

As 




Fig. 118 . 


Fig. 11] 


Fig.193 


Fig ug 


Fig. 19fi 




ic 

L^N 

\ 0<x \ 



5 f /) 



/c 


O 1 

C 

F ) 








\ 0 

a 

Fig.1919. 

J C 

\ .v' 

CL /) 

b \ \ 







rirr 

Fig. 194. / 

IV - -jD 

1 

t/ "v 


E 

\ 


I’/alc Will, fxuic /] 8. 


J.Uasirc sc. 

























































































4 





J li a sire .(•<*. 

































































































OP GENERAL PROPERTIES. 


179 


As the fedtions touch one another in d, by the pre- B O 

ceding Cor. they do not meet one another in three_ 

other points. Let the ftraight line 1 g therefore meet 
the curve of the fedtion xfjy'mf Then, by Cor. 2. 

Prop. XVII. Book I.be 2 :ge 2 ::ib x b f : f g x 
g 1 ; and be 2 :ge 2 ::ib xb/:/gxgi. Con- 
fequently (n.v.) ib xbfifg x gi: :ib x b/: 
f g x g 1, and by alternation ib xbf:ibxb/:: 
fgx G 1 : f g x G 1 ; and therefore (1. vi.) b f : 
b/::fg:/g. Hence (12. v.) bf : b/:: eg : bg, 
and therefore b f, b/ are equal : which is abfurd. 
Confequently the Cor. is evident. 

PROP. VIII. 

If a Jiraight line touch a conic fedion, and a ftraight line 
perpendicular to it be drawn through the point of con - 
tad, and meeting the axis or axes of the fedion, the 
fegment of the perpendicular between the point of con- 
tad and the axis of a parabola , or between the point of 
contad and the tranfverfe axis of the fedion, will be the 
leaft of all Jiraight lines which can be drawn from the 
fame point in the axis, and on the fame fide of it, to the 
curve; but the fegment of the perpendicular between the 
point of contad and the conjugate axis of the ellipfe will 
be the greatefl of all fraight lines which can be drawn 
from the fame point in the axis, and on the fame fide of 
it, to the curve . 

Let the ftraight line p r touch a conic fe&ion in the Fig. 140. 
point r, and let the ftraight line p k, perpendicular to 
the tangent, meet a b the axis of a parabola, or the 
tranfverfe axis of the fedfion, in the point k, and let it 
meet d e the conjugate axis of the ellipfe in m ; the 
fegment pk is the leaft of all the ftraight lines which 
can be drawn from k, on the fame fide of a b, to the 
n 2 curve; 


o> 



OF GENERAL PROPERTIES. 


18 o 

• BOOK curve; and in the ellipfe the fegment m p is the great- 
iV ' eft of all the ftraight lines which can be drawn from 
m, on the fame fide of d e, to the curve. 

Let the tangent p it meet the axis a b in r ; and 
draw p g a double ordinate to ar, and let it meet a b 
in F, and the curve again in g, and draw it g. Then, 
as the angles at f are right angles, and as p f is equal 
to f g, we have (4. i.) p r equal to g r, g k equal to 
p k, and k g r a right angle, being equal (8. i.) to the 
angle k p r. Let the ftraight line b h, touching the 
fedtion in the vertex b, meet the tangent p r in h, 
and draw K h. Then in the parabola the fquare of 
Fig. 140. B h is to the fquare of p h, as the parameter of the 
axis a b to the parameter of the diameter paffing 
through p, by Prop. IV. Book III. and therefore, by 
Cor. 1. Prop. XI. Book III. the fquare of bh is lefs 
than the fquare of p h. Confequently, as k h is com¬ 
mon to the two right angled triangles kbh, kph, 
the fquare of kb (47. i.) muft be greater than the 
fquare of k p, and K p muft be lefs than k b. Again, 
Fig. 141. in the ellipfe or hyperbola, c being the center, by 
I4 ~* Prop. V. Book II. the fquare of bh is to the fquare of 
p h as the fquare of c d to the fquare of the femidia- 
meter parallel to ph; and therefore, by Prop. XI. 
Book II. the fquare of b h is lefs than the fquare of 
p h. Confequently, for the fame reafons as in the pa- 
Fig. 140. rabola, pk is lefs than kb. If therefore with k as a 
* 4 *’ center, and k p as a diftance, in each fedtion, a circle 
be defcribed, its circumference will pafs through g, 
and cut a b within the fedtion * ; and as K p R, k g r 
are right angles, p r and gr (16. iii.) are tangents to 


* The circle is intentionally omitted in the figures. The defcrip- 
tion of it would have made them more complex, and not rendered the 
Propolition or either of the Corollaries more perfpicuous. 


the 



OP GENERAL PROPERTIES. 


181 


the circle, and, by Cor. 2. Prop. VI. they are alfo tan¬ 
gents to the feftion. Confequently, by Cor. 2. Prop. 
VII. the circumference of the circle cannot meet the 
curve of the fedtion in any other point befides p and G ; 
and therefore k p is the leaft of all ftraight lines which 
can be drawn from k, on the fame fide of a b, to the 
curve. 

In the ellipfe draw p n a double ordinate to the con¬ 
jugate axis d e, and let it meet d e in l, and the curve 
again in n. Let the tangent p r meet d e in t, 
and draw n t, n m. Then it may be proved, as above, 
that m n is equal to m p, n t equal to p t, and that 
the angle m n t is a right angle, being equal to the 
angle m pt. In the ellipfe let the ftraight line d y, 
touching the fedtion in the vertex d, meet the tangent 
p r in y, and draw m v. Then, by Cor. 3. Prop. III. 
Book II. d v is parallel to the axis a b, and by Prop. 
V. Book II. the fquare of d v is to the fquare of p v, 
as the fquare of c b to the fquare of the femidiameter 
parallel to p v ; and therefore, by Prop. XI. Book II. 
the fquare of d v is greater than the fquare of p v. 
Confequently, as the angles at d and p are right an¬ 
gles, and m v common to the two triangles m d y, 
m p v, the fquare of m p (47. i.) muft be greater than 
the fquare of md. If therefore with m as a center, 
and m p as a diftance, a circle be defcribed, its circum¬ 
ference will pafs through n, it will cut d e without 
the ellipfe, and, for the fame reafons as above, the 
ftraight lines t p, t n will touch it and the fedlion in 
the points p, n. Confequently, by Cor. 2. Prop. VII. 
the circumference of the circle cannot meet the curve 
of the fedtion in any other point befides p and n ; and 
therefore m p is the greateft of all ftraight lines which 
can be drawn from m, on the fame fide of d e, to the 
curve of the ellipfe. 

Cor . 1. If a ftraight line as pg be a double ordinate 
n 3 to 


BOOK 

IV. 


Fig. 141, 


182 


OF GENERAL PROPERTIES. 


BOOK to ab the axis of a parabola, or the tranfverfe axis of a 
IV * conic fe&ion, a circle touching the fe&ion in p, and 
’j, ig I4 „ paffing through g, will alio touch the fedtion in g ; 
141. and the other parts of its circumference will fall whol- 
I42 ‘ ly within the fedtion. For p G will be in the circle, 
and, being bifedted by a b at right angles, the center 
of the circle (Cor. 1. iii.) will be in a b. Let k be 
the center, and draw k p, and let p it be common to 
the circle and fedtion, according to the third Defini-* 
tion. Then (18. iii.) k p r is a right angle; and k g, 
g r being drawn, it may be proved, as above, that the 
angles k g r, k p r are equal. Confequently G r will 
touch the circle, (16. iii.) and it is evident, from Cor. 
2. Prop. VI. Book III. that it alfo touches the fedtion.' 
Hence the Cor. is manifeft. 

Fig. 141. Cor . 2. If a ftraight line as p n be a double ordinate 
I42# to d e the conjugate axis of an ellipfe, or of oppofite 
hyperbolas, a circle touching the ellipfe or hyperbola 
B p in p and paffing through n will alfo touch the el¬ 
lipfe or the oppofite hyperbola in n. For let the axis 
D e meet the common tangent p r in t, and the ordi¬ 
nate p n in l. Then, as p n will be in the circle, the 
center of the circle (Cor. 1. iii.) will be in d e. Let 
3u be the center, and draw m p, m n, n t. Then (4. i.) 
m p is equal to m n, and t p equal to t n ; and there¬ 
fore (8. i.) the angle m n t is equal to the angle m pt, 
which is a right one. Hence (16. iii.) the circle 
touches the ellipfe or the oppofite hyperbola in n, and 
n t is the common tangent to the circle and fe&ion. 

In this cafe it is evident, that the circle defcribed 
with the center m, and the diftance m p, falls without 
the ellipfe, and without each of the oppofite hyperbo¬ 
las. For, by this Prop, m p in the ellipfe is the great- 
eft ftraight line which can be drawn from m to the 
curve; and in the hyperbolas, as t p, t n are the com¬ 
mon tangents, it is evident that m p, m n are the lead 

ftraight 



OF GENERAL PROPERTIES. 


183 

ftraight lines which can be drawn from m to the op- book 
polite hyperbolas. . 


PROP. IX. 

Iffrom the vertex of the axis of a parabola , or from a 
vertex of the tranfverfe axis of an ellipfe or hyperbola , 
a fegment be taken in the axis equal to its parameter , a 
circle defcribed about this fegment as a diameter will 
fall wholly within the fcBion ; but if from a vertex of 
the conjugate axis of an ellipfe a fegment be taken in 
the axis equal to its parameter, a circle defcribed about 
this fegment as a diameter will fall wholly without the 
feftion. 

Let a b be an axis of a conic fe&ion, and in the hy- Fig. 143. 
perbola the tranfverfe axis, and from the vertex A let 
the fegment a c be taken in the axis equal to its para- J 4 6 * 
meter; the circle aec defcribed about a c as a dia¬ 
meter will fall wholly within the fe&ion, unlefs ab be 
the conjugate axis of the ellipfe, and if ab be the con¬ 
jugate axis of the ellipfe, the circle will fall wholly 
without the fe&ion. 

For through a draw the ftraight line A d equal to 
Ac, and at right angles to a b, and draw cd. Through 
any point Fin ac draw f g an ordinate to A b, and let 
it meet the circumference of the circle in e, the curve 
of the fe&ion in g, and the ftraight line c d in k. In 
the ellipfe and hyperbola draw from the vertex b the 
flraight line b d, and let it meet f g in h ; but in the 
parabola draw d h parallel to the axis a b, and let it 
meet f g in h. Then, by Prop. II. Book III. a d, 
f k are parallel, and therefore (4. vi.) ad : Ac: : fk : 
f c; and as A d is equal to ac, fk is equal to f c. 
Confequently a f x f k is equal to af x fc, and 
therefore (33. iii.) af x f k is equal to the fquare of 
N 4 EF, 



184 


OP GENERAL PROPERTIES. 


BOOK 

IV. 


Fig. H 7 - 

148. 

149. 


e f. But, by Cor. 1. Prop. VI. Book II. and Prop. Ill, 
Book III. the fquare of f g is equal to a f x f h. In 
the parabola and hyperbola therefore, and when a b is 
the tranfverfe axis of the ellipfe, the fquare of f g is 
greater than the fquare of f e, and confequently the 
point g is without the circle. But if a b be the con¬ 
jugate axis of the ellipfe, as in Fig. 146. the fquare of 
f e will be greater than the fquare of f g, and there¬ 
fore the point g will be within the circle. 

PROP. X. 

If from the vertex of the axis of a parabola, or from a ver¬ 
tex of the tranfverfe axis of an ellipfe or hyperbola , a 
fegment be taken in the axis greater than its parameter, 
a circle defcribed about this fegment as a diameter will 
fall without the fedion on each fde of the vertex ; but 
if from a vertex of the conjugate axis of an ellipfe a feg¬ 
ment be taken in the axis lefs than its parameter , a cir¬ 
cle defcribed about this fegment as a diameter will fall 
within the ellipfe on each fide of the vertex . 

Firft, let a b be the axis of a parabola, or the tranf¬ 
verfe axis of an ellipfe or hyperbola, and from the ver¬ 
tex a let the fegment a c be taken in the axis greater 
than its parameter; the circle aic defcribed about 
A c as a diameter will fall without the fe&ion on each 
fide of the vertex A. 

In the ellipfe let the point c be between the vertices 
A, b, and in each fection let the ftraight line ad be 
drawn perpendicular to the axis A b, and equal to its 
parameter. In a d let the fegment af be taken equal 
to a c, and draw c f. In the ellipfe and hyperbola let 
the ftraight line b d be drawn, and let it meet c f in 
E; but in the parabola let the ftraight line de be 
drawn parallel to the axis, and let it meet c f in e. In 

each 



I'lnli 1 1. it) / 



J. Dasire sc 



















































































































OV GENERAL PROPERTIES. ' 183 

each fe&ion let any point G be taken in c F, between BOOK 
the points e, f; and draw g l parallel to the ordinates IV * 
of a b, and let it meet d e in h, the circumference of 
the circle in 1, the curve of the fe&ion in k, and the 
axis a b in l. Then, by Prop. III. Book III. and Cor. 

I. Prop. VI. Book II. the fquare of the ordinate l k is- 
equal to the rectangle under al, l h ; and on account 
of the equals a f, a c, the redtangle- under a l, l g is 
equal to the re&angle under a l, l c, and therefore 
(35. iii.) equal to the fquare of 1 l. But the rcdtangle 
under a l, l g is greater than the redlangle under a l* 
l it, and therefore the fquare of 1 l is greater than the 
fquare of l k. Confequently the point 1 is without 
the fcdtion ; and if the flraight line G l meet the cir¬ 
cumference of the circle again in m, the point m, for 
the fame reafons as above, will be without the fedlion. 

The circle a 1 c therefore falls without the fedlion on 
each fide of the vertex a. 

Secondly, let ab be the conjugate axis of an ellipfe, Fig.150, 
and from the vertex a let the fegment a c be taken in 
the axis lefs than its parameter; the circle A 1 c de- 
fcribed about a c as a diameter will fall within the el¬ 
lipfe on each fide of the vertex a. 

For let a c be greater than the axis ab. Let the 
ftraight line A d be drawn perpendicular to A b, and 
equal to its parameter. Iika d let the fegment a f be 
taken equal to a c, and draw B d, c f, and let them 
meet one another in the point e. In c f take any 
point G between the points f, e, and draw G l parallel 
to the ordinates of a b, and let it meet b d in h, the 
curve of the ellipfe in k, the axis in l, and the cir¬ 
cumference of the circle in 1, m. Then, as above, it 
may be demonftrated, that the points 1, m are within 
the ellipfe ; and therefore that the circle a 1 c falls 
within the ellipfe on each fide of the vertex a. 


Cor. 



OF CIRCLES OF THE SAME 


186 

BOOK 

IV. 


Cor . r. If from the vertex of the axis of a parabola, 
or from a vertex of the tranfverfe axis of an hyperbola, 
or from a vertex of either axis in an ellipf’e, a fegment 
be taken in the axis equal to its parameter, a circle de- 
fcribed about this fegment as a diameter will have the 
fame curvature with the fedlion in the vertex. For, by 
the preceding Prop, in the parabola, and when A b is a 
tranfverfe axis, the circle aec falls wholly within the 
fedfion, the diameter a c being equal to the parameter 
of the axis A b ; and it is evident, that if a fegment be 
taken from a in a b lefs than a c, a circle defcribed 
about it as a diameter will fall within the circle aec. 
Again, by this Prop, if from a a fegment be taken in 
A b greater than the parameter of a b, or greater than 
A c, in Fig. 147. it will fall without the fedtion. In 
Fig. 143. therefore the circle aec has the fame cur¬ 
vature with the fedtion in A. Alfo when a b is the 
conjugate axis of the ellipfe, by the preceding Prop, 
the circle aec falls wholly without the ellipfe, A c 
being equal to the parameter of a b ; and it is evident, 
that if a fegment be taken from Ain ab greater than 
A c, the circle defcribed about it as a diameter will fall 
without the circle aec. Again, by this Prop, if from 
A a fegment be taken in a e lefs than the parameter of 
a b, or lefs than A c, in Fig. 150. a circle defcribed 
about it as a diameter will fall within the fedtion. In 
this cafe therefore the circle A e c in Fig. 146. has the 
fame curvature with the fedtion in A, according to the 
fourth Definition. 

Cor. 2. If a circle touch a conic fedtion in the vertex 
of an axis, and have the fame curvature with the fec¬ 
tion in the vertex, it will cut off from the axis a feg- 
ment equal to its parameter. This is evident from the 
preceding Corollary. 


PROP. ' 



CURVATURE WITH THE SECTIONS. 


187 


PROP. XI. 


BOOK 

IV. 


If from a point in the curve of a conic fedion a double or¬ 
dinate be drawn to the axis of a parabola , or to the 
tranfverfe axis of the fedion , and if through the point 
in which it meets the curve again a diameter be drawn , 
and from the jirft mentioned point a double ordinate be 
drawn to this diameter ; a circle touching the fedion in 
the frjl mentioned point , and puffing through the other 
extremity of the lajl mentioned double ordinate , will not 
meet the fedion in any other point befdes thefe two , and 
it will have the fame curvature with the fedion in the 
point of contad . 


From the point b in the curve of the conic fedion Fig. 153. 
B A l let the ftraight line b a be drawn a double ordi- 
nate to d h the axis of the parabola b a l, or to d h 156. 
the tranfverfe axis of the fedion, and from a, the point 
in which it meets the curve again, draw the diameter 
a e, and from b draw the double ordinate b f to a e 5 
the circle b k f, touching the fedion in b and palling 
through f, the point in which the double ordinate b f 
meets the curve again, will not meet the fedion in any 
other point belides b, f, and it will have the fame 
curvature with the fedion in the point b. 

For if the circle k f b pafs through the point A, it 
will touch the fedion alfo in a, by Cor. 1. Prop. VIII. 
and the other parts of the circumference will fall whol¬ 
ly within the fedion ; and if the fedion be an ellipfe, 
and b e be drawn, it will be an ordinate to the conju¬ 
gate axis ; and therefore if the circle k f b pafs 
through e, it will alfo touch the fedion in e, by Cor. 

2 . Prop. VIII. and the other parts of the circumfe¬ 
rence will fall wholly without the ellipfe. If there¬ 
fore the circle k f b pafs through a in any fedion, or 

through 



ids 

BOOK 

IV. 


OF CIRCLES OF THE SAME 

through e in the ellipfe, it will touch the fe£lion in 
the points b, a or b, e, and, by Cor. 2. Prop. VII, it 
will not pafs through f, contrary to hypothecs. Hence 
it is evident that the circle kb f in any fe&iori is 
greater than a circle which touches the le&ion in 
B, A, but in the ellipfe it muft be lefs than a circle 
which touches the ellipfe in b, e ; and therefore, 
in any fe£tion, the circumference of the circle k b f 
meets the ftraight line b a without, but the ftraight 
line b e within the fe£tion. Let the circumference 
meet b a in k, and A e in q. If it be poffible, let 
the circumference of the circle meet the feclion 
in l* Draw the common tangent b d, meeting the 
axis d h in d, and draw a d. Then, by Cor. 2. 
Prop. VI. Book III. a d will touch the feftion, and 
therefore, by Prop. II. Book III. a d, b f are parallel, 
and on account of the axis d h, the tangents (4. i.) 
A d, b d are equal. Let l m be drawn parallel to ad 
or f b, meeting the curve again in 1, and the tangent 
B d in m. Then, by Prop. XIII. Book I.bd 2 : ad 2 :: 
b M 2 : l m x mi; and therefore b m 2 is equal to l m 
x mi; and as, by hypothefis, the point l is in the 
circumference of the circle, the point 1 (36. iii.) is alfo 
in the circumference of the circle. The circle k f b 
therefore, touching the ledtion in b, meets the fedtion 
in f, l, 1, which, by Cor. 1. Prop. VII. is impofiible. 
The circle kfb therefore does not meet the fedtion in 
any point befides b, f ; and as the point k is without, 
and the point a within the fedtion, the arch b k f will 
be without, and the arch f q. b will be within the fee- 
tion. 

The circle k fb will alfo have the fame curvature 
with the fedtion in the point b. For let any other cir¬ 
cle as b r l, touching the fedtion in b, be deferibed ; 
and, firft, let b it l be lefs than k f b. Then, as the 

ftraight 



CURVATURE WITH THE SECTIONS. 1 % 

ftraight line b d touches the two circles b r l, ic F b book 
in the fame point b, the centers of thefe circles (18. * v * 

iii.) will be in the fame ftraight line, and therefore the Fig ^ 
lefier circle b r l will fall wholly within the circle 154- 
kfb. The circle b r l therefore cannot pafs between 
the arch b q. f and the curve of the fedlion ; and if it 
pafs through a, or be lefs than a circle palling through 
a, it will fall wholly within the fedlion, by Cor. 1. 

Prop. VIII. and therefore it cannot pafs between the 
arch F K b and the curve of the fedlion. But let the 
circle b r l be greater than the circle paffing through 
A, and meet the ftraight line B a in v, and the curve 
of the fedlion in l*. Let the ftraight line l m be drawn 
parallel to A d, meeting the curve of the fedlion again 
in 1, and the tangent b d in m. Then, by Prop. XIII, 

Book I. b d 2 : A d 2 : : b m 2 : l m x m i, and, on ac¬ 
count of the equals b d, ad, the fquare of b m is equal 
to lm x m 1. The point 1 therefore (36. iii.) is in 
the circumference of the circle, and confequently, by 
Cor. 1. Prop. VII. it meets the fedlion only in the 
points B, l, 1. Again, it is evident that the circle 
b r l meets b f within, and b a without the fedlion, 
and therefore that the curve of the fedlion between the 
points f, l are without the circle. In the tangent 
b m therefore take any point t between b and m, and 
let the ftraight line t p be drawn parallel to a d, meet¬ 
ing the fedlion in p, s , and the circle b r l in n > p. 

Then, as above, it may be demonftrated that b t 2 is 
equal to p t x t s. But the fquare of b t (36. iii.) is 
equal to the redlangle r t x t p, and therefore the 
re&angles p T x rt x t/> are equal.. Confe¬ 
quently and as p t is greater 

than t r, t p is greater than t 5. The arch Bp 1 there¬ 
fore falls within the fedlion. 

* Such lines as * could eafily be fupplied by the mind of the reader 
are omitted in Fig. 155. and 156, * 

Se- 



190 


OF CIRCLES OF THE SAME 


BOOK Secondly, let the circle brl be greater than the 
IV * circle kfb; and it may then be demonftrated, as 
Figr I55> above, that the circle brl falls without k f b, and 
therefore that it cannot pafs between the arch f k b 
and the fedtion. Again, if the circle brl pafs through 
33 in the elliple, or be greater than the circle paffing 
through e, it will fall wholly without the ellipfe, and 
therefore it will not pafs between the ellipfe and the 
arch fob. But in any fedtion let the circle brl 
meet a e within the fedtion, and the curve of the fec- 
tion in l. Draw the flraight line L m parallel to A d, 
and let it meet the fedtion in 1, and the tangent bd in 
M. In b d take any point t between b and m. Draw 
T r, and let it meet the fedtion in p and j, and the cir¬ 
cle b r L in r and p. Then, as above, it may be de¬ 
monftrated that the arch p 1 is without the fedtion. 
The circle kfb therefore has the fame curvature with 
the fedtion in the point of contadt b, according to 
Def. IV. 

Cor . From the above, and Cor. 1. Prop. X. it is evi¬ 
dent that only one circle, touching a conic fedtion in a 
given point, can have the fame curvature with the fec- 
tion in that point. 

PROP. XII. 

If a circle touch a conic fedtion ^ and have the fame curva¬ 
ture with the fedtion in the point of contact , it will cut 
off from the diameter of the fedtion paffing through the 
point of contact a fegment equal to its parameter. 

If the point of contadt be the vertex of an axis, the 
Propofition has been demonftrated, as hated in Cor. 2. 

Fig. \ 6 o. Prop. X. but if the circle kbf touch the fedtion in the 
162! P 0 * 111 ^ B > which is not a vertex of an axis, let every thing 
remain as in the preceding Prop, and let k b f be the 
circle having the fame curvature with the fedtion in 

the I 





CURVATURE WITH THE SECTIONS. 

the point B. Let the circumference of the circle kbf 
therefore meet b c, drawn through b the point of con¬ 
tact and c the center, in the point n, if the fe&ion be 
an ellipfe or hyperbola; or let it meet the diameter 
B n in the point n, if the fe&ion be a parabola; in ei¬ 
ther cafe the fegment b n is equal to the parameter of 
the diameter bcn. 

Firft, let the fe&ion be an ellipfe or hyperbola, and 
draw the diameter c v parallel to the tangent d a, and 
the diameter t m parallel to the tangent bd, and meet¬ 
ing b f in g. Let the tangent d a meet the diameter 
T m in m, ,and the diameter bc in l. Let the diame¬ 
ter a e meet its ordinate b f in i. Then, on account 
of the axis d c and its ordinate a b, the lemidiameters 
' c A, c b are equal; and by fimilar triangles la : ac : : 
2i ; ci; and a c : a >i : : c i : I G. Confequently, 

L A : A c : A M 
B i : c i : i g, 

and ( 22 . v.) L A : A M : : b 1 : 1 G ; and therefore ( 22 . 
vi.) l A x a m : B 1 x 1 g : : l a 2 : b i 2 , or, by the 
above, as A c 2 to c i 2 . But, by Cor. 2. Prop. IV. 
Book II. the diameters t m, b l are conjugate, and 
therefore l a x a m is equal to c v 2 , by Cor. 2. Prop. 
IX. Book II. Confequently cv 2 :bixig::ac 2 : 
c 1 2 ; and, by alternation, c v 2 : A c a : : b 1 x 1 g : 
ci 2 ; and, by Prop. V. Book II. c v 2 : a c 2 : : b i 2 ; 
A 1 x ie. Confequently, by the tenth Lemma, (and 
12 . v. and 3. ii.) c v 2 : a c 2 : : G b x bi : Ac 1 , and 
therefore (14. v.) c v 2 is equal to g b x b i. But, by 
Prop. V. Book II. a d 2 : bd 2 : : c v 2 : c T 2 ; and there¬ 
fore, on account of the equals a d, e d, the fquare of 
c v is equal to the fquare of c t. The fquare of c t 
is therefore equal to g b x b i ; and n b x b c is 
equal'to fb x b g, by the feventh Lemma. If there¬ 
fore n B be bifedted in p, the rectangle under p b, b c 

will 


191 


BOOK 

IV. 


Fig. 160. 
16 



OP CIRCLES OP THE SAME 

BOOK will be equal to g b x b i, or to c t 2 . But c t 2 is 
IV * equal to the reCtangle under c b and half the paratne- 
ter of c b. Confequently the fegment bn is equal to 
the parameter of the diameter c B. 

Fig. 162. Secondly, let the fe&ion be a parabola, and let the 
diameter a e meet its ordinate b f in e. Draw to the 
diameter B n the ordinate a c, meeting the diameter , 
B n in c, and bf in h. Then, on account of the equals 
A d, b d, the parameter of the diameters a e, b c will 
be equal, by Prop..IV. Book III. and by Cor. Prop. V. 
Book III. a e, B c are equal, and as ae,bc are paral¬ 
lel, the triangles a e h, b c h are equiangular. Con¬ 
fequently (4. vi.) ae:eh::bc:bh, and therefore 
(14. v.) e h, h B are equal. Confequently fb:be:j 
b e : b h, and therefore f b x b h is equal to e e 2 . 
But, by the feventli Lemma, fb X b h is equal to n b 
Xbc; and as be,ac are equal, e e 2 is equal to a c 2 . 
The reCtangle n b c therefore is equal to the fquare of 
AC. Moreover the fquare of a c is equal to the rect¬ 
angle under b c, and the parameter of the diameter 
b n ; and therefore the reCtangle n b c is equal to the ; 
reCtangle under the abfcifs b c, and the parameter of 
the diameter b n. The fegment b n is therefore equal 
to the parameter of the diameter drawn through b the 
point of contaCt. 

Cor. 1. If from the vertex of a diameter of a para¬ 
bola, or from a vertex of a tranfverfe diameter of an hy¬ 
perbola, or from a vertex of any diameter of an ellipfe, 
a fegment be taken in the diameter equal to its para¬ 
meter, a circle touching the feCtion in the vertex, and 
palling through the other extremity of the fegment, 
will have the fame curvature with the feClion in the 
vertex. This is evident from Cor. 1. Prop. X. and the 
above. 

Cor. 2. If through 0, the focus of the parabola, the 

ftraight 




CURVATURE WITH THE SECTIONS* 


*93 

ftraight line b r be drawn, and meet the circle of cur- BOOK 
vature again in r; b r will be equal to b n, the para- IV ‘ 
meter of the diameter b c. 

For through o draw o x parallel to b d, and let it 
meet b c in x. Then, by Cor. Prop. XII. Book III. 

B x is equal to bo j and by the feventh Lemma r b 
X bo is equal to N b x b x. Confequently b r is 
equal to b n. 

Cor. 3. The reft remaining as above in the ellipfe and Fig. 1 60. 
hyperbola, if the ftraight line b r drawn through the 
focus o meet the circle again in r, and x s be the tranf- 
verfe axis, then b r will be to the diameter c t as c T 
is to x s. For let the diameter ct meet b r in y, and 
then, by the feventh Lemma, the rectangle .rby is 
equal to the rectangle f b g. But by the above the 
re&angle peg is equal to twice the fquare of c T, and 
therefore the re&angle r b y is equal to twice the fquare 
of c t. Confequently r b : the whole diameter ct:: 
the femidiameter c t : b y. But, by Cor. Prop. XVI. 

Book II. b y is equal to cx, and therefore (15. v.) br : 
the whole diameter ct:: the whole diameter c x : x s. 

PROP. XIII. 

If a circle touching an ellipfe or hyperbola have the fame 
curvature with the feCtion in the point of contact , its fe- 
midiameter will be to the femidiameter of the feCtion con¬ 
jugate to that paffing through the point of contad, as the 
fquare of the fame femidiameter of thefedion to the rect¬ 
angle under the femiaxes • 

% 

For if the circle touch the fe<$lion in the vertex of Fig. T44. 
an axis, then every thing remaining as in Prop. IX. 
by the Definition of a parameter, and inverfion, a c is 
to the axis parallel to the common tangent at a, as the 
fame axis to the axis ab. Confequently (15. v. and 

o i.iv.) 



194 


OF CIRCLES OF THE SAME 


BOOK 

IV. 


Fig. 160. 
161. 


I. iv.) as the femidiameter of the circle aec to the 
femiaxis parallel to the common tangent at a, fo is 
the fquare of the fame femiaxis to the redhangle under 
the femiaxes. 

But if the circle of curvature do not touch the fee- 
tion in the vertex of an axis, let every thing remain as 
in Prop. XII. and let u b be - the femidiameter of the 
circle; and then u b will be to c t as c T 2 to the 
redlangle under the femiaxes. 

For let c x be the tranfverfe femiaxis, and c h the 
conjugate femiaxis. From the center c draw the per¬ 
pendicular c q to the tangent b d. Let u b meet the 
circumference again in w, and draw wn. Then (31. 
iii.) the angle w n b is equal to the angle c a b, and' 
the angle new (29. i.) equal to the angle bcq. 
Confequently (4. vi.) c B : c a : : w B : b n, or (15. v.) 
as u b to p u ; and therefore c a X u B is equal to 
c b x p b. But, by Prop. XII. and the Definition of 
a parameter, c b X p b is equal to c x 2 , and therefore 
u b : c t : : c t : c Q. Confequently (1. vi.) u b : 
ct : : ct 2 : ct xca. But, by Cor. 1. Prop. XIX. 
Book II. c t X c a is equal to c x X c h •, and 
therefore ub:ct::ct 2 :cx X ch. 

Cor. By the above the fquare of c x is equal to the 
redtangle under u b, c a. 

PROP. XIV. 

If from the center of a circle , touching a conic feBion, and 
having the fame curvature with the feBion in the point 
of contaB , a perpendicular he let fall upon a flraight 
line drawn from the point of contaB through the nearejl 
focus , a flraight line drawn from the point of CGncourfe 
to the point in which the diameter of the circle, paffing 
through the point of contaB, cuts the focal axis will he 
at right angles fo this diameter of the circle . 


From 



CURVATURE WITH THE SECTIONS. 


*95 


BOOK 

IV. 


Fig. 157. 
158. 
* 59 ' 


Fig. 1 ^ 7 - 
158. 


From the center of the circle lpm, touching 
the conic fection a p in the point p and having the 
fame curvature with the fe&ion in p, let the perpendi¬ 
cular h k be drawn to the ftraight line p k palling 
through f the neareft focus; the ftraight line k i, 
drawn from k the point of concourfe to 1 the point in 
which the diameter p H of the circle cuts the focal axis 
A 1, i& at right angles to h p. 

For firft let the fe&ion be an ellipfe or hyperbola of 
which c is the center, a c the tranfverfe femiaxis, and 
d c the conjugate femiaxis. Let n p be the common 
tangent, and c t the femidiameter of the fe&ion pa¬ 
rallel to np; and draw c G, f n perpendicular to 
N p. Then, by Cor. Prop. XIII. the lquare of c t is 
equal to the re&angle under h p, c g ; and, by Prop. 

XVII. Book II. the fquare of c d is equal to the reft- ' 
angle under 1 p, c g. Confequently c t 2 :c d 2 : : 
hp x c G : 1 p x cg; and therefore (t. vi.) c t 2 : 
c D 2 : : h p : 1 p. But, by Prop. XIX. Book II. (and 
22. vi.) ct 2 :cd 2 ::fp 2 :fn 2 ; and, on account of 
the limilar triangles, f p 2 : f n 2 : : h p 2 : p k 2 . Con¬ 
fequently (11. v.) h p 2 : p k 2 : : h p : 1 p; and there¬ 
fore (1. vi.) h p 2 : p k 2 : : H p 2 : H p X I P, and (14. v.) 
p K 2 is equal to h p X 1 P. Hence 11 p : p k : ; p K : 

1 p, and (6. vi.) p i k is a right angle. 

Secondly, let the fe&ion be a parabola, and let the Fig. 159. 
common tangent n p meet the axis a i in g. Then, 
by Cor. Prop. IX. Book III. G f is equal to f p, and 
therefore (6. i.) the angle f g p is equal to the angle 
fpg. But as each of the angles gpi, h k p is a right 
angle, the angles fpg,fpi together are equal to the 
angles k p i, p h k together. Confequently the angle 
I g p is equal to the angle phk; and therefore (q.vi.) 
gi:ip::pk:ph, and h p x p i is equal to g 1 x 
pk. But, by Cor. 2. Prop. XII. (and 3. ii.) p k is 
o 2 equal 



196 OF CENTRIPETAL FORCED 

book equal to half the parameter of the diameter palling 

IV * _through p, and therefore, by Prop. XI. Book III. p k 

is equal to g i. Confequently h p x p i is equal to 
p k 2 . As above, therefore, hp:pk::pk:ip, and 
( 5 . vi.) p i k is a right angle. 

Fig. 157. Cor. If a ftraight line n g touch a conic fe&ion in p, 
and p h at right angles to it meet the focal axis a b in 
1, and if 1 k perpendicular to p h meet in the point k 
the ftraight line p l drawn through the neareft focus 
f, and, laftly, if k h at right angles to p l meet p h 
in h, the point h will be the center of the circle which 
touches the feftion, and has the fame curvature with 
it in p. 

SCHOLIUM. 

Fig. 160. If a body b revolve in afpace void of reliftance about 
the center c in the curve b a f, and if the circle b k f 
touch the curve in the point b, and have the fame cur¬ 
vature with it in that point; and if the ftraight line 
B c n meet the circle again in n, and c a be perpendi¬ 
cular to the common tangent b q, Sir Ilaac Newton 
has demonftrated, in Cor. 3. to Prop. VI. Lib. I. of the 
Prineipia, that the centripetal force is reciprocally as 

c a 2 x n b, or dire&Iy as-^--. 

C G x N B 

By means of this expreflion, and the properties of 
ofculating circles demonftrated in the preceding Pro- 
politions, the centripal forces of bodies moving in co¬ 
nic fedtions may be eafily afcertained, as in the follow¬ 
ing examples. 

Fig. 160. Let the body revolve in the ellipfe x H s, and let 
the law of centripetal force tending to c the center be 
required. 

Every thing remaining as in Prop. XIII. the centri¬ 
petal force, according to the Newtonian expreflion, is 
reciprocally as c a 2 x b n. But, by Cor. 1. Prop. 

XIX. 




OF CENTRIPETAL FORCES. 


I 9 7 


XIX. Book II. ct x ca = xcx ch, and therefore BOOK 

X C 1 x C H 2 

c a 2 =--—3 and by the Definition of a parame--* 


ter, and Prop. XIII. b n = 


4 CT 
acE 


2 CT 
C B 


. Confequent- 


X C X C H 


2 C T 


XC 1 X 2 CH i 


ly c a 2 x b n = 

CT C B CB 

and therefore, as x c 2 x 2 c h 2 is a conftant quantity, 
the centripetal force is reciprocally as , or dire&ly as 
the diftance c b. 

2. Let a body p move in an ellipfe or hyperbola p A, 
and let the centripetal force tending to the focus f of 
the fe&ion be required. 

The reft remaining as in Prop. XIV. let the ftraight Fig. 157. 
line p f meet the circle again in l, and, by Cpr. 3. 158 * 

Prop. XII. lp:2ct: :2ct:2ca j and there¬ 
fore l p = • Again, by Prop. XIX. Book 

II. (and 22. vi.) fn 2 : fp 2 : : cd 2 : ct 1 , and f n 2 = 

p p 1 ^ C D 2 

- pp -. But, the Newtonian general expreflion 

being adapted to the prefent Figures, the centripetal 
force is reciprocally as fn 2 x l p ; and therefore this 

F P a X C D 2 

force, by the above, is reciprocally as 

2 C T z FP 2 XKD" 


C T 


Again, if the letter l be put for 
c a c a & r 

the parameter of the tranfverfe axis, 2 a c : 2 c d : : 


2 C B : 


4 C D 


2 C D 


= l. Confequently as l, or its 

value, is conftant, the centripetal force is reciprocally 
as f p% or inverfely as the fquare of the diftance. 

3. Let a body p move in the curve of a parabola pa, 
and let the law of centripetal force tending to the fo¬ 
cus f be required. 

03 


Every 















i 9 8 


OF STRAIGHT LINES 


book Every thing remaining as in Prop. XIV. let the 
1V> flraight line p f meet the circle again in l, and let f n 

Fig. 1 59. be drawn perpendicular to the tangent p g. Then, by 
Cor. 2. Prop. XII. of this, and Cor. 2. Prop. XI. Book 
III. l p = 4 f p; and, by Prop. X. Book III. fn^fp 
x a f. Again by the Newtonian general expreffion, 
adapted to this Figure, the centripetal force is recipro¬ 
cally as f n 2 x l p. By the above therefore the cen¬ 
tripetal force is reciprocally asFPXAFX4FP = FP J 
x 4 a f. Confeqnently, as 4 a f is conftant, the cen¬ 
tripetal force is reciprocally as f p 2 . 

PROP. XV. 

If three flraight lines touch a conic fedion , or oppoflte hy¬ 
perbolas, any one of them will he harmonically divided 
in its point of contad, the points in which it meets the 
other two , and the points in which it meets the flraight 
line joining their points of contad . 

Fig. 15Let the three ftraight lines a r, r p, p n touch a 
conic fe&ion a e n, or oppofite hyperbolas q, n in the 
*66. points a, e, n ; any one of them as rp is harmoni¬ 
cally divided in its point of contact e, the points R, p 
in which it meets the other tangents, and the point A 
in which it meets the line a n joining their points of 
conta&. 

Fig. j<t. Cafe 1. Firfl, let the tangents q r, n p be parallel; 
I5 *‘ and then, by the Cor. to Prop. XIII. Book I. re: 
e p : : a r : n p. But (4. vi.) qr:np::ra:ap, 
and therefore (n. v.) r a : a p : : r e : e p. 

Fig. 165. Cafe 2. Let the tangents a r, n p meet one another 
l 6 6 6 j in the point b and through the point p draw p h pa¬ 
rallel to a r, and let it meet the curve of the fe&ion 
in g, h, and a n in 1. Then, by Prop. XIII. Book I. 
r E a : e pV: : a r 2 : h p x pg. But, by Prop. XVII. 
Book I. H r X p G is equal to p i 2 . Confequently R e 2 : 



CUT HARMONICALLY BY THE SECTIONS. 


I99 


E p 2 : : a r 2 : p 1% and (22. vi.) re:ep::gr:pi, BOOK 
But (4. vi.) a r : p 1 : : r a : a p ; and therefore (11. IV * 
v.) R a : A p : : R E : E p. 

PROP. XVI. 

If two Jlraight lines touching a conic feElion, or oppojite 
hyperbolas, meet one another, a fccant puffing through 
the point of concourfe will be harmonically divided in 
the point of concourfe, the points in which it meets the 
curve or curves, and the point in which it meets the 
Jlraight line joining the points of contaEl. 

Let the two ftraight lines e a, fa, touching the fee- Fig* 151* 
tion e d f, or the oppofite hyperbolas e, f, in the 
points e, f, meet one another in a, and let the ftraight 
line a e meet the curve or curves in b, d, and the 
ftraight line e f in c ; the ftraight line A b is harmoni¬ 
cally divided in the points a, d, c, b. 

For through b, d draw the ftraight lines g m, h k 
parallel to e f, and let them meet the tangents e a , 
f a in g, m, and h, k and the curve or curves in b , 
l , and d , 1. Then, by Prop. IV. g e, l m are equal, 
and therefore g l is equal to b m ; alfo h d, i k are 
equal, and therefore h 1 is equal to d k. By equian¬ 
gular triangles gb is to h d as a b to a d, and b m, 
or its equal g l, is to d k, or its equal H 1, in the fame 
proportion. Confequently (ii.v.)gb:hd::lg: 

1 h, and by alternation gb:lg::hd:ih; and 
therefore (22. vi.) b g x g l : d h x h 1 : : g b 2 ; 

H d 2 . But, on account of the parallels, g b 2 : h d 2 : : 
ab 1 : ad 2 ; and, by Prop. XIII. Book I. b g x g l : 
d h x h i : : e g 2 : e h 2 . All'o, on account of the pa¬ 
rallels, (10. vi.) e g 2 : e h 2 : : c b 2 : c D 2 •, and there¬ 
fore (11. v.) a b 2 : a d 2 : : c B 2 : c D 2 . Confequently 
(22. vi.) A B : A D : : c b : c d. 

If a b bife£l e f in c, by Cor. 1. to Prop. V. Book 
o 4 III, 


/ 



200 


OF STRAIGHT LINES 


BOOK HI. it will be a diameter. If therefore in this cafe A b 
* meet the curve of the fe&ion in two points, or the 
curve of each of the oppofite hyperbolas in one, it mud 
be a diameter of an ellipfe, or a tranfverfe diameter of 
an hyperbola; as a diameter of a parabola can meet 
the curve in one point only, and in the hyperbola a fe- 
cond diameter ; does not meet either of the oppofite 
curves. Confequently, by Cor. 3. Prop. VII. Book II. 

A b : ad;:bc:cd. 

PROP. XVII. 

If two flraight lines touching a conic feftion, or oppojlte ] 
hyperbolas , meet one another ,v and a fecant pafs through 
the point of concourfe , tangents pajjing through the points 
in which the fecant meets the curve or curves will either :■ 
he parallel , or they will meet one another in the Jlraight 
line joining the points of contact of the two firfl men~ 
tioned tangents . 

Fig. 170. Let the ftraight lines K l, k m, touching the conic 
fe£tion l g M, or the oppofite hyperbolas l, m in the 
173. points l, My meet one another in k, and let the ftraight 
line K b meet the curve, or oppofite curves, in the 
points g, b; ftraight lines touching the curve or curves 
in g and b 1 will either be parallel, or they will meet in 
the ftraight line q m. 

For if the ftraight line ke bifeft l m, by Cor. 1. 
Prop. VI. Book III. k b is a diameter, and l m is a dou¬ 
ble ordinate-to it; and, by Prop. II. Book III. tan¬ 
gents palling through g, b will be parallel to l m, and 
therefore parallel to one another. But let k b meet 
l m in h, and not bifect it. Let the tangents gn,bn 
be drawn, and meet one another in n, and if it be pof- 
fible let n not be in the ftraight line l m. Draw n l, 
and let it meet k b in v. Let the tangents n g, n b 
meet the tangent k 1 in s and t. Then, by Prop. XV, 




PhtU \\/ /'iiiir !/'<> 


Fig. 144. 




H 

7 l 1 ’ 

E]W 

c 


\ 


B 

\ 




7 Z 

V 

I<\ N 

V H /‘J 


vy l 


IUn. 

\ C 

/lyj N 


















































































































JR/itr. .XHLpagt wo, 


Fig. ISO 


J. Basin sc. 
























































































CUT HARMONICALLY BY THE SECTIONS. 


201 


t k : k s : : t l : l s ; and therefore, on account of BOOK 
the harmonicals n k, n g, n v, n b, by the twelfth IV ' 
Lemma, bk:kg::bv:vg. But, by Prop. XVI. 
bk:kg::bh:hg; and therefore (it. v.) b v : 

V G : : b h : h G. Confequently (17. and 18. v.) b g : 
v G : : b g : h g, and (14. v.) vg is equal to h g ; 
which is abfurd. The tangents g n, b n meet there¬ 
fore in the ftraight line l m. 

Cor. 1. If a ftraight line as l m, not palling through 
the center, cut a conic fe&ion, or oppofite hyperbolas, 
in l, m, and from points n, d, &c. in l m, tangents 
n b, n g, d e, d f, &c. be drawn to the curve, or op¬ 
pofite curves, the ftraight lines bg, fe, &c. each join¬ 
ing the points of contact of two tangents drawn from 
the fame point in l m, will meet one another in the 
point k, in which the tangents palling through l, m 
meet one another. 

Cor. 2. If the ftraight lines bg, f e, not palling 
through the center, meet the curve, or oppofite curves, 
in b, g and f, e, and one another in k, and if they be 
harmonically divided in b, h, g, k and f, q, e, k ; 
then tangents palling through b, g, or through f, e, 
will meet one another in the ftraight line drawn through 
H, Q. 

Cor. 3. If the tangents b n, g n meet in N, and the 
tangents f d, e d in d, and the ftraight lines bg,fe 
joining the points of contaft meet one another in k, 
and the ftraight line n d in h and a; the ftraight lines 
kb, k f will be harmonically divided in K, g, h, b 
and in k, e, a, f. 

PROP. XVIII. 

Jf the oppofite Jides of a trapezium inf crib ed in a conic Jec- 
tion be not parallel , and neither pafs through the center , 
the interfeElion of ftraight lines joining the oppofte an- 
gular points , the inter fettion of two of the oppofte fdes, 

and 



so z 


OF STRAIGHT LINES, &C. 


BOOK 

IV. 


Fig, 164. 


and the interjections fituated between thefe Jides , of tan • 
gents poffing through the points in which thefe fules 
meet the curve or curves , will he all four in the fame 
Jlraight line . 

Let g b f e be a trapezium infcribed in a conic fec^ 
tion, or oppofite hyperbolas, having no two of its oppo- 
fite fides parallel, and no one of them palling through the 
center ; the interfe&ion 1 of the ftraight lines b e, f g 
joining the oppofite angular points, the interfe£tion of 
the oppofite fides b f, g' e, and the interfe&ions d, n, 
fituated between g v, b v, of the tangents e d, f d, 
and g n, b n, are all four in the fame ftraight line. 

Let the fides b g, f e meet one another in k 5 and 
N d being drawn, let it meet k b in m, k f in l, and 
B f in v. Then, by Cor. 3. Prop. XVII. k b is harmo¬ 
nically divided in k, g, m, b, and k f in k, e, l, f. 
Tirft, if the ftraight line ge do not pafs through v, 
draw v k, v e ; and let v e meet k b in q. Then, on 
account of the harmonicals v k, v a, v m, vb, the 
ftraight line K b will be harmonically divided in k, q, 
m, b ; which is abfurd, by Cor. 3. to the firft Defini¬ 
tion before Lemma XI. Confequently bf, ge meet 
one another in the ftraight line n d. Secondly, if n d 
do not pafs through the point 1, draw 1 k, m i, and 
let m 1 meet k f in p. Then, on account of the har¬ 
monicals 1 k, G f, m p, be, the ftraight line kf is 
harmonically divided, by Lemma XII. in k, e, p, f ; 
which, by Cor. 3. to the firft Definition, before Lemma 
XI. is abfurd. Confequently the ftraight lines B e, g f 
meet one another in the ftraight line n d ; and 1, v, b, 
n the points of the interfe&ions are in the fame ftraight 
line n d. 

SCHOLIUM. 

In the following Problems, when the expreffion Art.' 

with 



SOLUTIONS OP PROBLEMS, 


203 


with a figure occurs, the article fo numbered in the BOOK 
Scholium at the end of the third Book is referred to, 1V * 

The fir ft fix of the following folutions, although in 
feveral refpedts different, apply to the 59th, 60th, and 
61 ft Problems in the Arithmetica Univerfalis, and alfo 
to the 32nd, 33rd, 24th, 25th, 26th, and 27th propo- 
fitions in the firft Book of the Principia. 

PROP. XIX. PR OB. I. 

Given Jive points in the curve of a conic feStion , to de¬ 
fer ibe the Jeftion. 

Let e. A, b, c, d be five points given in the curve pig, 
of a conic fed! ion, to deferibe the fedtion. 

Draw a b, e d, d c, c A, b c ; and through the 
point e draw e l parallel to A c, and e t parallel to 
a b. Let the ftraight line e l meet e c in l, and the 
ftraight line D c in h ; and let e t meet b d in n. In 
e t, and on the faijie fide of eh with n, take the 
fegment et fo that h e may be to e n as l k to e t. 

Then bt being drawn it will touch the fedtion, by 
Cor. 3. Prop. V. In the fame way ftraight lines d f, 
a g may be drawn touching the fedtion in d, a. Then 
if any two of the tangents, fuppofe d f, b t be paral¬ 
lel, the fedtion will be an ellipfe, according to Prop. 

VIII. Book I. and d b will be a diameter according to 
Cor. 1. Prop. II. Book II. In this cafe if a ftraight 
line be drawn from e parallel to d f, or b t, it will be 
an ordinate to b d ; and the conjugate diameter to 
e d being found by art. 10. the fedlion may be de- 
feribed. 

But if no two of the tangents be parallel to one 
another, let b t meet d f in f, and a g in g. Draw 
F k bifedting b d in k, and g i bifedling ab in 1. 

Then, by Cor. 1. Prop, VI. Book III. f k, g i will be 

dia- 



204 


SOLUTIONS OP PROBLEMS. 


BOOK 

IV. 


Fig. 176. 
*77. 


17 *. 


v 




diameters, and therefore if they are parallel, the fedtion 
will be a parabola : but if they are not parallel, they 
will meet in the center of the fedtion. 

If the fedtion be a parabola let FK.be bifedted in p, 
and then it is evident, by Prop. V. Book III. and art. 9. 
that the fedtion may be defcribed. But if the fee- 
lion be not a parabola, let f k, gi meet in o, and o 
will be the center. Between f o, o k let a mean pro¬ 
portional o p be found, and, by Prop. VII. Book II. 
op will be a femidiameter of the fedtion, to which 
b d is a double ordinate. Confequently the diameter 
conjugate to op being found by art 10. the feclion 
may be defcribed. 

PROP. XX. 

If the Jlraight lines AC, B D cutting a conic feftion in 
A, c and B, d, and meeting one another in G, meet in 
T and f a Jlraight line T f which touches the feftion 
in E, the fquare of et will he to the fquare of ef in 1 
the ratio compounded of the ratio of the rectangle under 
B G, g D to the re £1 angle under A G, G c and of the ra¬ 
tio of the reftangle under at, t c, to the reftangle un¬ 
der B F, F D. k 

Fir ft let the fedlion be a parabola or hyperbola, 
and let the point t be on that fide of the figure on 
which the fedtion can be extended. Let the ftraight 
line 1 k be drawn parallel to b d, and let it meet the 
curve in 1, k ; and let the fquare of the ftraight line 
v x be equal to the reftangle 1 t k. Then, by Prop. 
XIII. Book I. the fquare of e t is to the fquare of ef 
as the fquare of v x to the redtangle bfd. But the 
fquare of v x is to the redtangle bfd in the ratio 
compounded of the fquare of v x to the reftangle 
at c and of the ratio of a t c to the reftangle bfd,; 








Plate JUU1I. page ic.f- 



J.Basire sc 

























































































SOLUTIONS GF PROBLEMS; 

that Is, by Prop. XIII. Book I. in the ratio com¬ 
pounded of the ratio of the redangle bgd to the 
redangle agc, and of the ratio of the redangle a t c 
to the redangle b f d. The fquare of e t therefore 
(n. v.) is to the fquare of ef in the ratio com¬ 
pounded of the ratio of the redangle bgd to the 
redangle agc, and of the ratio of the redangle atc 
to the redangle bpd, 

Secondly, let the fedion be an ellipfe, of which o is 
the center, and, the reft being as above, let o a be the 
femidiameter parallel to b d, o d the femidiameter pa¬ 
rallel f t, and o b the femidiameter parallel to a c. 
Let it be v x 2 : a t X t c : : o a % : o b z ; and then, 
as by Prop. V. Book II. a t x t c : e t 4 : : o b z : 
o d z y we have 

y x 2 : A t x t c r E t 4 
O a 1 : o b z : o d z . 

Confequently (22. v.) v x 2 : e t 4 : : o a % : O d 1 ; and 
therefore by Prop. V. Book II. (and n. v.) vx 2 : et 2 : : 
B f x f d : e f 2 , and by alternation e t 2 : e f 2 : : v x 2 : 
b f x f d. Alfo, by the above, and Prop. V. Book II. 
(and 11. v.) v x 2 : A T X tc : b G x G d : a g x g c. 
Again, the fquare of v x is to the redangle under b f, 
f d in the ratio compounded of the ratio of the fquare 
of v x to the redangle under a t, t c, that is, by the 
above, of the redangle under bg, g d to the redangle 
under a g, g c, and of the ratio of the redangle under 
A t, t c to the redangle under b f, f d. The fquare 
of e t therefore (11. v.) is to the fquare of e f in the 
ratio compounded of the ratio of the redangle under 
B g, g d to the redangle under a g, g c and of the 
ratio of the redangle under A t, t c to the redangle 
under bf, f d. 

Cor . 1. The next remaining as above, if the ftraight 
line t F meet in the point h the ftraight line u l, 

which 


205 

BOOK 

IV. 


Fi£. 


Fig. 176. 
177* 



Zo6 SOLUTIONS OB* PROBLEMS# 

BOOK which touches the fection in the point r and meets 
IV * t c in L, it may be demonftrated in the fame manner 
that the fquare of e t is to the fquare of E h in the 
ratio compounded of the ratio of the fquare of p l to 
the rectangle under A L, lc, and of the ratio of the 
reCtangle under a t, t c to the fquare of h p. 

Cor. 2 . If the fquares of the flraight lines m n* 
Rs^tg, yz be equal to the rectangles b g x g d, 
ag x g c, A t X TCj b f x F d, each to each ; then 
E t will be to e f as the rectangle under m n, t a to 
the reCtangle under y z, r s. For, by the above, the 
reCtangle under ag, g c is to the reCtangle under b g^ 
g d as the reCtangle under A t, t c to the fquare of v x 5 
and therefore as the fquare of r s is to the fquare of 
m n, fo is the fquare of a t to the fquare of v x. Con- 
fequently (22. vi.) as R s is to m n, fo is t q to v x, 
and therefore (16. vi.) the reCtangle under r s, v x is 
equal to the reCtangle under M n, t a. Again, from 
the above, as the fquare of e t is to the fqnare of e f, 
fo is the fquare of v x to the reCtangle under bf,fd, 
or the fquare of yz; and therefore, as e t to ef 
fo is v x to y z, that is (1. vi.) the reCtangle under 
v x, r s to the reCtangle under y z, rs. Confe- 
quently, on account of the equal reCtangles, e t is to 
e f as the reCtangle under m n, t a to the reCtangle 
under y z, rs. 

Cor. 3. Hence if any two flraight lines a c, b d 
cut a conic feCtion in a, c and e, d, and meet one an¬ 
other in g, and meet in t and f the flraight line t f, 
which touches the feCtion, the point of contaCt may 
be found. For let e be fuppofed to be the point of 
contaCt, and as above e t is to e f as the reCtangle 
under a mean proportional between r g, g d and a 
mean proportional between at, tc to a reCtangle 
under a mean proportional between b f, f d and a 


mean 



SOLUTIONS OF PROBLEMS. $Of 

mean proportional between ag, g c. In the fame book 
way the point e may be found, if the tangent t f 
meet in h the ftraight line h l, which touches the ~" 

fettion in p, and meets in l the ftraight line A c, which 
cuts the fettion in a, c, and meets t f in t. 

PROP. XXL P ROB. II. 

Four points in the curve of a conic fettion,'and a ftraight 
line touching the fettion, being given , let 'it be required 
to defcribe the fettion . 

Let the four points A, b, d, c in the curve of a conic Fig. 176. 
fettion, and the ftraight line t f touching the fettion, 
be given in pofition ; it is required to defcribe the fee- 379* 
tion. 

Cafe 1. Let the tangent t f pafs through the point 
B, and draw a b, b d, d c, c a. Firft, let the oppofite 
iides A c, e d of the trapezium be parallel, and let k r 17$. 
be drawn bifetting b d in k, and a c in r. Then k r 
is a diameter, by Cor. 2. Prop. II. Book III. and if it is 
parallel to the tangent f t, it will be the conjugate di¬ 
ameter to b d, by Cor. 2. Prop. IV. Book II. and the 
point K will be the center of the fettion. Confequent- 
ly, if from the point a a ftraight line be drawn ordi- 
nately applied to the diameter b d, the magnitude of 
the diameter k. r will be determined, by art. 10. and 
the fettion may then be deferibed, by art. 8 . But 
if the diameter k r be not parallel to the tangent f t, 
let them meet in f. Draw F D, and it will touch the 
fettion, by Cor. 2. Prop. VI. Book III. Through c 
draw the ftraight line c 1 parallel to the tangent f t, 
and let it meet the tangent f d in e, and b d in g. 

Let the fegment e i be fo taken in“ c g, that c e may 
be to e g as e g to e i ; and when the point G is with¬ 
out the fettion, let the point c be between e and 1. 

Then 



SOLUTIONS OF PROBLEMS* 


20 S 

BOOK Then the point 1 will be in the curve of the fecHorn,- 
1V ‘ by Prop. XVII. Book I. and the five points a, b, d, c, i 
being in the curve, the fe&ion may be defcribed as 
in Prop. XIX. 

Secondly* let the oppofite fides of the trapezium be 

Pig- 179* not parallel* and let ab, c d meet in l. Draw A d* 
c b, and let them meet in p. Through the points p* l 
draw the ftraight line p l* and let it meet the tangent 
t F in F, and draw f £>. Then f d will touch the fec- 
tion* by Prop. XVIII. and, as above* a fifth point I 
may be found in the curve. 

Tig. 176. Cafe 2. If the tangent tf do not pafs through a 
given point, the point of contadl may be found, by Cor. 
3. Prop. XX. and the fe&ion may be defcribed as in 
Prop. XIX. 


PROP. XXII. PROB. HI. 

Three points being given in the curve of a conic feBiori, 
and two ftraight lines touching it being given in pofttioriy 
let it be required to defcribe the feftion . 

Cafe 1. If each of the two tangents pafs through 
one of the given points, the points of contact will be 
given, and a ftraight line drawn through the third 
given point in the curve, and parallel to the ftraight 
line joining the points of contact, will meet the tan¬ 
gents. The fegments of this ftraight line between the , 
given point in the curve and the tangents will there¬ 
fore be given ; and if they are unequal* the line will be 
a fecant, and the other point in which it cuts the curve 
may readily be obtained, according to Prop. IV. The 
folution may then be completed* by Prop. XXI. But 
if the fegments are equal, the line will be a tangent, 
and in this cafe call it the third tangent* and draw a 
ftraight line from its point pf contact to the point of 



SOLUTIONS OF PROELEMS. 


S09 


contact of the firft tangent. Draw a ftraight line from 
the point in which the firft and third tangents meet to 
the point of contact of the fecond tangent. Then, by 
Prop. XVI. this ftraight line will be harmonically di¬ 
vided in the point in which the firft and third tangents 
meet, the point in which it meets the ftraight line 
joining their points of contact, the point of contact of 
the fecond tangent, and the other point in which it 
meets the curve. This other point may therefore be 
found by the fecond and third Corollaries in page 155, 
and the folution may be completed as above. 

Cafe 3. Let the firft tangent pals through one of the 
given points, but let the fecond tangent not pafs through 
either of the other two; and, firft, let the two tangents 
be parallel. Through the two other points draw a 
ftraight line, and if it be parallel to the tangents, a 
ftraight line drawn from the point of contact of the 
firft tangent and bife<fting this fecant will meet the fe¬ 
cond tangent in its point of contact, as is evident from 
Prop. I. Book II. But if the ftraight line palling 
through the other two given points be not parallel to 
the tangents, it will meet them, and then, by Prop. 
XIII. Book I. the recftangle under its fegments be¬ 
tween the points and the firft tangent will be to the 
fquare of the firft tangent, as the re&angle under its 
fegments between the given points and the fecond tan¬ 
gent to the fquare of the fecond tangent. The point 
of contact of the fecond tangent will therefore be ob¬ 
tained. Secondly, let d g be the firft tangent palling 
through jb one of the given points, and let a, c be the 
other two; and let d k the other tangent not be pa¬ 
rallel to d g, but let them meet in d. Draw a c, and 
let it meet ok in k. Then if a c be parallel to one 
of the tangents, as in Fig. 180. let k e be a mean pro¬ 
portional between c k, k a, the point e being in k c ; 

p and 


BOOK 

IV. 


Fig. 1^0. 
181. 



210 


SOLUTIONS OF PROBLEMS. 


BOOK 

IV. 


Tig. 182. 


and a flraight line drawn through b and £ will meet d k 
in its point of conta6l, by Prop. XVII. Book I. But 
if the flraight line A c be not parallel to either tangent, 
as in Fig. 181. let it meet d g in G, and d f in k ; 
and/ let g k be fo divided, that the re&angle under 
A k, k c may be to the rectangle under cg, g a as the 
fquare of k e to the fquare of eg. Draw b e, and it 
will meet the tangent d f in the point of contaft, by 
Cor. 2. Prop. XVII. Book I. 

Cafe 3. Let neither of the tangents k f or D G pafs 
through a given point, and let a, b, c be the given 
points. Draw a c, and let it meet the tangent d g in 
d, and the tangent kf in k. Let k d be fo divided 
in e, that the re&angle under a k, k c may be to the 
re&angle under c d, d a as the fquare of k e to the 
fquare of e d, and the flraight line joining the points 
of conta6t will pafs through e, by Cor. 2. Prop. XVII. 
Book I. Again, draw c b, and let it meet the tangent 
d g in l, and the tangent kf in 1. Let l 1 be fo di¬ 
vided in m, that the re6langle under cl, lb may be 
to the re&angle under bi, 1 c as the fquare of l m to 
the fquare of m i, and the flraight line joining the 
points of contact will pafs through m, by the fame as 
above. Confequently the flraight line m e will meet 
the tangents in the points of conta£l. 

In every cafe, therefore, a fe£tion may be defcribed, 
by Prop. XIX. or Prop. XXI. 

Cor. If two tangents be given in pofition, and alfo 
two points in the curve of a conic fection, but without 
the tangents, the point may be found in which the fe- 
cant, palling through the given points, meets the 
flraight line joining the points of contact, provided the 
fecant be not parallel to the ftraight line joining the 
points of contact. For if the tangents and fecant be 
parallel to one another, the point in which the fecant 

is 



SOLUTIONS OF PROBLEMS# 


air 


is bife&ed will be the point required, as ftated in the book 
fecond cafe. In other cafes the Cor. is evident from n ‘ 
the above; for the tangents d g, d k being given in 
pofition, and the fecant pafling through the given 
points a, c, as in Fig. 180, 181, 182. the point e was 
afcertained. 


PROP. XXIII. 

If three Jlraight lines touch a conic JeBion , a Jlraight line 
drawn through the point in which the JirJl and fecond 
tangents meet one another will he harmonically divided 
in this point of concourfe, in the point in which it meets 
the third tangenty and in the points in which it meets 
Jlraight lines drawn from the firf andfecond points of 
contact through the third point of contabi . 

Let the ftraight lines e f, e g, g h touch the conic Fi S- 
fe6lion in the points f, c, h, the ftraight line e n, 
drawn through the point e, in which the firft tangent 
E f and the fecond e g meet one another, is harmoni¬ 
cally divided in e, in the point n in which it meets 
the third tangent g h, and in the points p, n in which 
it meets the ftraight lines f h, c h, drawn from the 
firft and fecond points of contact f, c through the third 
point of conta& h. 

For let e n meet the curve of the fe£tion in the 
points s, i; and then, by Cor. 2. Prop. XVII. Book I, 
sexei:srxsi : : e p 2 : p R 2 : : e n 2 : n r 2 . 
Confequently (22. L)ep:pr::en:nr. 

PROP. XXIV. PR OB. IV. 

Two points being given in the curve of a conic feftion , and 
three Jlraight lines being given in pojition and touching 
the curve , let it be required to dejcribe the fettion, 

P 3 


The 



11 % 

BOOK 

IV. 


Fig. 183. 


Fig. 184. 
185. 

Fig. 184. 


SOLUTIONS OF PROBLEMS* 

The two points A, b being given in the curve of a 
conic fedtion, and the three ftraight. lines c d, e f, g h 
being given in pofition and touching the fedlion, let it 
be required to defcribe the fedtion. 

Cafe 1 . Let c d, g h two of the given tangents pafs 
through the two given points a, b, and let them meet 
the other given tangent ef in d and g. If the tan¬ 
gent e f be parallel to a b, the ftraight line joining the 
points of contadt, and g d be bifedled in e, the point 
e will be that in which g d touches the feclion, by 
Prop. IV. But if g Dj a b be not parallel, let them 
meet in f, and let g d be fo divided that d f may be 
to f g as d e to e g, and e will be the point in which 
g d touches the fedtion, by Prop. XV. Three points 
will therefore be obtained in the curve, and confequent- 
ly the folution may be completed, by Prop. XXII. 

Cafe 2. Let g h one of the given tangents pafs through 
b one of the given points, and meet the given tangent 
e f in G. Let the ftraight line a b be drawn, and let 
it meet the given tangent c d in d ; and if a b be pa¬ 
rallel to e f, let the fegment d l be taken in b d a 
mean proportional between b d, da, and the ftraight 
line joining the points in which e f, c d touch the 
fedtion, will pafs through the point l, by Prop. XVII. 
Book I. Draw g l, and let it meet the tangent c D 
in k ; and if the tangents g h, c d be parallel, let l g 
be to l k as b g to k c, and c will be the point in 
>vhich c d touches the fedtion. For let g k meet the 
curve of the fedtion in p and n, and the redtangle un¬ 
der p G, g n is to the redtangle under n k, k p as the 
fquare of g l to the fquare of l k, by Prop. XVII. 
Book I. and the fquare of b g is to the fquare of c p, 
the fegment between the point k and the point of con- 
tadf, in the fame ratio, by Prop. XIII. Book I. If the 
ftraight line a b be not parallel to the tangent e f, the 

point 



SOLUTIONS OF PROBLEMS. 


point l may be found by Cor. 2. Prop. XVII. Book I. BOOK 
and the tangents G h, c d being parallel, the point of IV ‘ 
contad c may be found as above. 

But if the tangent g h meet the tangent e f in g, Fig. 185. 
and the tangent CD in h, let g h be fo divided in n, 
that h b may be to b g as h n to g n, and the ftraight 
line joining the points of contad of the tangents e f, 
c d will pafs through n, by Prop. XV. Again, let the 
ftraight line a b meet the tangent e f in p, and the 
tangent c d in », and as the rectangle under a p, p b 
is to the redangle under b d, d a, fo let the fquare of 
p l be to the fquare of d l, and the ftraight line join¬ 
ing the points of contad of the tangents e f, c d will 
pal’s through l, by Cor. 2. Prop. XVII. Book I. Let 
the ftraight line l n therefore be drawn, and it will 
meet the tangents e f, c d in the points of contad. 

If the ftraight line a b be parallel to the tangent f g, 
the point l may be found, by Prop. XVII. Book I. as 
above. 

Cafe 3. Let the points a, b be without any one of Fig. 18 6 . 
the tangents, and let the ftraight lines ab,ef,gh be 
/parallel. Let the tangent cd meet the tangent g ii 
in g, and the tangent e f in e ; and let the ftraight 
line a b meet the tangent c d in d. In a d let the leg- 
ment d l be taken a mean proportional between a d, 
d b, and the ftraight line joining the points of contad 
, of the tangents c d, e f will pafs through the point l, 
by Prop. XVII. Book I. Let the ftraight line gl n 
be drawn; let it meet the tangent ? f in k, and let g l 
be to l k as g n to n k, and the ftraight line joining • 
the points of contad of the tangents g h, e f will pals 
through n, by Prop. XXIII. Again, let a b be bi- 
feded in p, and the ftraight line joining the points of 
contad of the tangents g h, e f will pafs through r, 
by Prop. I. Book II. Confequently if the ftraight line 
p 3 N P 



SOLUTIONS OF PROBLEMS. 


2x4 

BOOK 

IV. 

Fig. 187. 


Fig. 188. 


n p be drawn, it will meet the tangents in the points of 
contaft. 

Cafe 4. Let the tangent c d meet the tangent g h 
in g and e f in e, and let the ftraight line a b be pa¬ 
rallel to the tangent c d, and meet the tangent gh in 
k, and the tangent e f in m. In the ftraight line k M 
let the fegment m p be taken a mean proportional be¬ 
tween a m, m b ; and let the fegment k l be a mean 
proportional between bk ; ka; and the ftraight line 
joining the points of contact of the tangents cd, ef 
will pafs through the point p, but the ftraight line 
joining the points of contact of the tangents c d, g h 
will pafs through the point l, by Prop. XVII. Book I. 
Let the ftraight line g p be drawn, and meet the tan¬ 
gent e f in 1, and as g p to p 1 fo let g n be to n i. 
Again, let l e be drawn, and let it meet the tangent 
g h in q, and let a l be to l e as ft r to r e. Then, 
by Prop. XXIII. the ftraight line n r pafles through 
the points of conta& of the tangents EF, g h. 

Cafe 5. Let the ftraight line ab be not parallel to 
any one of the tangents g h, d c, e f, and let it meet 
the tangents DC, ef in e, the point in which they 
meet one another. Let the ftraight line a b be divided 
in the point l, fo that b e may be to e a as b l to 
l a, and the ftraight line joining the points of contact 
of the tangents dc,ef will pafs through the point l, 
by Prop. XVI. Let the ftraight line gl be drawn, 
and meet the tangent e f in k, and in the ftraight line 
G l let the fegment g n be fo taken, that g l may be 
to LK as GN to nk3 and, by Prop. XXIII. the ftraight 
line joining the points of contact of the tangents e f, 
gh pafles through the point n. Again, let the ftraight 
line a b meet the tangent g h in m, and in a b let the 
fegment a r be fo taken that the redtangle a e b may 
be to the redtangle a m b as the fquare of e r to the 

fquare 



SOLUTIONS OP PROBLEMS# 21 $ 

fquare of r m, and the ftraight line joining the points O K 
of contact of the tangents ef, gh will pafs through 
the point r, by Prop. XVII. Book L Confequently 
the ftraight line n r will meet the tangents ef ; gh 
in the points of contact. 

Cafe 6. Let the tangent c D meet the tangent g h Fig. 189, 
in g, and the tangent e f in e. Let the ftraight line 
a b be drawn, and let it meet the tangent g h in m, 
the tangent e f in k, and the tangent c d in d. Let 
the ftraight line a b be fo divided in l and n, that the 
rectangle under a k, k b may be to the rectangle un¬ 
der b m, m A as the fquare of k l to the fquare of ml; 
and as the rectangle under a d, d b to the rectangle 
under b m, m a, fo let the fquare of D n be to the 
fquare of nm. Then the ftraight line joining the 
points of contaft of the tangents ef,gh will pafs 
through the point l, but the ftraight line joining the 
points of conta& of the tangents cd 5 gh will pafs 
through n, by Prop. XVII. Book I. Let the ftraight 
line N e be drawn, and meet the tangent g h in r ; 
and let n e be fo divided in p, that e n may be to n r 
asEPtoPR; and, by Prop. XXIII. the ftraight line 
joining the points of contact of the tangents e f, g h 
pafles through p. Confequently if the ftraight line 
l p be drawn, it will meet the tangents e f, g h in the 
points of contact. 

In every cafe therefore the fe&ion may be defcribed 
by Prop. XIX. as five points may be eafily found. 

PROP. XXV. 

If the four Jlraight lines A e, e g, g h, h d touch a co - Fig. 174. 
nic fe&ion in A, b, c, d, and meet one another in E, G^ 

F, h ; and if the Jlraight lines AC, bd he drawn join¬ 
ing the oppojite points A, c and B, D and meeting one 
another \ the Jlraight line F g drawn through the oppo - 
p 4 jite 



21 6 
book 

IV. 


SOLUTIONS OF FIIOBLEMS. 

fite points in which the tangents meet one another wilt 
pafs through the point in which the Jlraight lines ac, 
b d meet one another . 

For let the ftraight line f g meet the fe&ion or op- 
pofite fe£tions in n, m and the ftraight line ac in i, 
and, by Cor. 2. Prop. XVII. Book I. the re&angle un¬ 
der n g, G„ m is to the rectangle under n f, f m as the 
fquare of g 1 to the fquare of f 1. But, if it be poffi- 
ble, let the ftraight line f g meet the ftraight line b d 
in p, and, as above, the rectangle under n g, g m is to 
the rectangle under nf,f m as the fquare of p g to 
the fquare of f p. Confequently (11. v.) as the fquare 
of g 1 to the fquare of f i, fo is the fquare of pg to 
the fquare of f p ; and therefore (22. vi.) as g 1 to f 1 
fo is p g to f p. Confequently (18. v.) as gi to f g 
fo is g p to f G, and therefore (14. v.) the ftraight 
lines gi, gp are equal $ which is abfurd. 

The reft remaining, if the ftraight line f g be a dia¬ 
meter of a parabola, or parallel to an afymptote of an 
hyperbola, the fquare of n i, and alfo the fquare of 
n p, will be equal to the re&angle under g n, n f, by 
Prop. XXIII. Book III. and therefore gi, g p are 
equal; which is abfurd. 

In any cafe therefore the ftraight lines a c, b d, f g 
meet one another in 1, and if the ftraight line e h be 
drawn joining the remaining oppofite points, in which 
the tangents meet one another, it will pafs through 
the point 1, for the fame reafons as above. 

Cor . Hence, if four ftraight lines ae,eo,gh,hd 
touch a conic fe&ion and meet one other in e, g, f, h, 
and if the ftraight lines f g, e h joining the oppofite 
points meet one another in 1 ; the ftraight lines a c, 
b d joining the oppofite points of contact will pafs 
through the point 1. 


PROP. 



SOLUTIONS OF PROBLEMS* 


PROP. XXVI. PROS. V. 

The point A being given In the curve of a conic fe&lon } , 
and the four ftraight lines e f, e g, g h, h d touching 
the feEllon being given in pojitlon , let It be required to 
defcrlbe the feEllon . 

Let the tangents meet one another in e, f, g, h, 
and let the ftraight lines f g, eh be drawn,joining 
the oppofite points f, g and e, h, and meeting one 
another in i. Let the ftraight line A i be drawn, and 
let it meet the oppofite tangent g h in c, and if the 
point a be in one of the tangents, the ftraight line gh 
will touch the fe&ion in c, as is evident from Prop. 
XXV. But if the point a be not in one of the tan¬ 
gents, let a c meet e f in k, and let the ftraight line 
K c be fo divided in 11 that the fquare of k i may be 
to the fquare of i c as the rectangle a k r to the re 6 t- 
angle rca; and as by the laft Prop, the ftraight line 
joining the points of contaft of e f, g h paftes through 
i, the point r will be in the curve of the fe£tion, by 
Cor. 2. Prop. XVII. Book I. Confequently in any cafe 
the feftion may be deferibed, by Prop. XXIV. 

PROP. XXVII. PROB. VI. 

Five flralght lines being given In pofition and touching a 
conic feEllon , let it be required to find the points In which 
they touch the feEllon. 

Let the ftraight lines ae,bc,cd,de, e a touch a 
conic fe&ion, and let it be required to find the points 
of contact in them. 

Let a B c d E a be the quinquelateral figure con¬ 
tained by the tangents, and let a b be called the firft 
fide, b c the fecond, &c. and let fbcd be the qua¬ 
drilateral figure contained under the four firft fides, and 
draw the diagonals b d, f c meeting one another in m. 


217 

BOOK 

IV. 

Fig. iH. 
* 75 - 


Fig. 19c, 



SOLUTIONS OF PROBLEMS. 


2lS 


BOOK 

IV. 


Fig. 19I. 


The firft fide ab of the quinquelateral figure being 
now omitted, let 1 c d e be a quadrilateral contained 
by the others, and let 1 d, c e the diagonals be drawn 
meeting one another in n. Then m n being drawn, it 
will pals through the points in which the fecond fide 
bc and the fourth de touch the fe&ion, by Cor. Prop. 
XXV. In the fame manner the points may be found 
in which ab, cd, ae touch the fe&ion, and there¬ 
fore the fe&ion may be defcribed, by Prop. XIX. 

PROP. XXVIII. 

Let ED be an equilateral hyperbola , of which AF, Ac 
arc the ajymptotes , and let it cut in the point D the curve 
of the parabola A d, of which. A F is the axis, and the 
J'egment A f equal to the parameter of the axis ; let there 
be drawn to the curve of the hyperbola the fraight line 
F E parallel to the afympiote A c, andfrom the point D, 
hi which the curves of the hyperbola and parabola cut 
one another , let there be drawn to the afymptote A f the 
Jlraight line D b parallel to the afymptote a c; then 
will the fraight lines ed,AB be two mean propor¬ 
tionals between af, f e. 

For as e d is an equilateral hyperbola, the angle 
A fe is a right one, by Prop. XVI. Book III. (and 
29. i.) The ftraight line d b is therefore an ordinate 
to the axis of the parabola, and, by Prop. III. Book III. 
(and 17. vi.) af:bd::bd:ab. Again, by Cor. 2. 
Prop. XVII. Book III. a f : a b : ; b d : f e, and 
therefore by alternation af:bd::ab:fe. Con- 
lequently (11. v.) a f : b d : : b d : a b, and b d : a b 
: : a b : f e. 

Cor. Hence if two ftraight lines as af, fe be given, 
two mean proportionals may be found between them. 
For let the two ftraight lines a f, f e be at right an- 



f 


Platt X!UV.f>age ’.iff 




































































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. 
















ft 



























































































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* 

















■ 









































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SOLUTIONS OP PROBLEMS. 


319 


gles to one another, and let the parallelogram afeg 
be completed. Let the parabola a d be defcribed, of 
which a F is the axis, and the fegment a f equal to its 
parameter. Again, let an equilateral hyperbola be de- 
fcribed through the point e, of which a f, a g are the 
afymptotes, and let its curve cut the curve of the pa¬ 
rabola in d. Let D b be drawn to a f and parallel to 
a g, and let d c be drawn to a g parallel to af. Then 
the ftraight lines b d, a b will be two mean propor¬ 
tionals between af, f e. 


PROP. XXIX. 

Let A E be a parabola, of which A D is the axis , and A B 
a fegment in it equal to half its parameter; let the 
Jlraight line B G be perpendicular to the axis, and draw 
A G ; with the center G and dijlance G A defcribe the 
circle ace cutting the axis in the point c and the curve 
of the parabola in E, and let ed^ drawn an ordinate 
to the axis; the Jlraight lines E D, A D will be two 
mean proportionals between A c and a Jlraight line equal 
to the double of G B. 


For, by the conftruction, (and 3. iii.) the ftraight 
line a c is equal to the parameter of the axis, and there¬ 
fore, by Prop. III. Book III. the fquare of d e is equal 
to the rectangle under a c, a d. Let d e meet the 
circumference of the circle again in f, and let the feg¬ 
ment f h be equal to the fegment d e. Then the rect¬ 
angle e d f will be equal to the reCtangle a d c, 
(35. iii.) and therefore the fquare of d e together with 
the reCtangle e d f are equal to the rectangles d a c, 
adc together, that is, (2. ii.) to the fquare of a d. 
But the fquare of n E together with the re&angle 
E d f is equal to the reft angle under d e, and a ftraight 
line equal to the fum of e d, d f (i. ii.) ; and there- 


BOOK 

IV. 


Fig. 19 s, 



SOLUTIONS OF PROBLEMS. 


ROOK fore the fquare of de together with the redangle ed? 

_is equal to the redangle under d e, h d. Confequent- 

ly (17. vi.) de:ad::ad:hd; and by the above 
A c : d e : : d e : a d. But dh is double of g b ; for 
let g 1 be drawn parallel to a d, and let it meet d h in 
1. Then g b, i d (34. i.) are equal to one another, as 
are alfo (3. iii.) e i, i f to one another, and therefore 
h 1 is equal to 1 d. The Propofition is therefore evi¬ 
dent. 

Cor. Hence, by means of a parabola and a circle, a 
method is evident of finding two mean proportionals 
between two given ftraight lines. 

PROP. XXX. 

Fig. 193. From any point B In the curve of the equilateral hyperbola 
B e let the Jlraight lines BA, bd he drawn to the 
afymptotes c A, CD, and let & A be parallel to C D and 
B D parallel to c A, and let A D the diameter of the pa¬ 
rallelogram be drawn ; with the center B and a diftance 
equal to the double of a d let a circle be defcribed , and 
let it meet the curve of the hyperbola in E’jfrom e draw 
E f to the afymptote c d and parallel /oca; then , A f 
being drawn , the angle baf will be a third part of 
the angle bad. 

For let a f meet b d in g. Bifed d f in k, and 
draw k 1 parallel to b d, and let it meet a f in the point 
1. Draw d 1. Then as the hyperbola is equilateral, 
the angle acd is a right one, and therefore (29. i.) 
each of the angles fki, d k i is a right one, and 
(4. i.) f 1, 1 d are equal. But, on account of the equals 
f k, k d and the parallels k i, d g, f i is equal to 1 g. 
Again, (15. and 29. i.) the triangles abg,fca are 
equiangular, and therefore (4. vi.) ab:bg:;cf: 
c a, and (1 6 . vi.) the redangle under b g, c f is equal 

to 



SOLUTIONS OF PROBLEMS, 


221 


to tlie redangle under b a, a c. But, by Prop. XVII. 
Book III. the redangle under b a, a c is equal to the 
redangle under e f, c f, and therefore the redangle 
under b g, c f is equal to the redangle under e f, c f. 
Confequently b g is equal to ef, and therefore (33. i.) 
B e is equal to g f; and therefore, by the conltrudion, 
F 1, 1 d, d a are equal. The angles dfi, f d i are 
therefore equal to one another, as are alfo the angles 
D 1 a, d a 1 to one another. The angle d a i is there¬ 
fore equal to the double (32. i.) of d f I, or of it9 
equal the angle bag. Confequently the angle bag 
is equal to a third part of the angle bad. 

Cor. Hence, by means of an equilateral hyperbola 
and its afymptotes, an angle may be divided into three 
equal parts. 


BOOK 

IV. 


A TREATISE 







' 

--S'. 

«£ C 3 - V* . *. 

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1 ( 1 ; ‘ 1 i - Cp ,;. 


♦ .••;• w ■ ' : . ’ • -1 • ^ . * $ V '. 1 . ‘ \ 

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s'odrar/ff f, ~: - > : to . •■-■’-■ • 

* • >tiJ v *i . ■ • i ».,* Jv; V \ 






















































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A 

TREATISE 


ON THE 

PRIMARY PROPERTIES 

OF 


CONCHOIDS, THE CISSOID, THE QUADRATRIX, 
CYCLOIDS, THE LOGARITHMIC CURVE, 

AND 

THE LOGARITHMIC, ARCHIMEDEAN, AND 
HYPERBOLIC SPIRALS. 







« 


















i i 4 


* 













9 












'** • 

♦ • r'**’ i 1 




% 












I 

X 


> 




































/ 




























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« 




V 




A 


TREATISE 

ON THE 

PRIMARY PROPERTIES 

OF 

CONCHOIDS, THE CISSOID, THE QUADRATRIX, 
CYCLOIDS, THE LOGARITHMIC CURVE, 

AND THE 

LOGARITHMIC, ARCHIMEDEAN, AND HYPERBOLIC SPIRALS, 


UU- ' 

SECTION I. 

Of Conchoids , 


DEFINITIONS, 

I. 

Xf the fixed point p be without the ftraight linePlateXXVZ 0 
T R x, and if the ftraight line dl of indefinite length Flg * 
pafs through p, and d, a be two fixed points in D L; 
then if the ftraight line d l, always palling through p, 
be moved in fuch a manner that the point a is always 
in t r x, and the point d defcribe the curve d vc ; the 
curve D v g is called a Conchoid. 

II. 

The fixed point p is called the Pole or Center of the 
conchoid, and the ftraight line t R x is called the Di- 
rettrix of the conchoid. 


Q 


III. 








OF CONCHOIDS* 


SECT. III. 

The ftraight line prv perpendicular to the directrix'* 
meeting it in r and the curve in v, is called the 
Axis of the conchoid ; and the point v is called the 
Vertex of the axis. 

IV. 

If the dire&rix be between the curve and the pole, 
as in Fig. 1. the curve is called the Superior Conchoid , 
or the Conchoid of Nichomedes. 

V. 

If the curve be between the directrix and the pole, 
and the fegment r v of the axis be lefs than the feg- 
ment p r, as in Fig. 2. the curve is called the Inferior 

Conchoid . 

VI. 

If the curve and pole be on the fame fide of the di¬ 
rectrix, and the fegment v r of the axis be greater than 
the fegment p r, as in Fig. 3. the curve d p y g is 
called the Nodaled Conchoid . 

Corollary to the preceding Definitions . In any one of 
the three conchoids if a ftraight line pafs through the 
pole, and cut the direCtrix and curve, its fegment in¬ 
tercepted between the direCtrix and the curve will be 
equal to the fegment r v of the axis, between the di¬ 
rectrix and curve. For the points d, a being fixed in 
the ftraight line d l, the magnitude of the fegment 
D A is the fame in every pofition of the moving line 
d l 5 and when d l falls upon the axis the defcribing 
point d coincides with y, and the point a coincides 
with r. 

VII. 

A ftraight line drawn from any point in the con¬ 
choid perpendicular to the axis is called an Ordinate to 
the axis. 


PROP. 



OP CONCHOIDS. 


PROP. I. 

The conchoid and its dire&rix being produced , on either 
Jide of the axis , continually approach nearer and nearer 
to one another , hut never meet. 

For, the reft remaining as in the Definitions of each 
of the conchoids, let the ftraight line d b be perpendi¬ 
cular to the direCtrix t x. Then p k, d b being per¬ 
pendicular to the direCtrix t x, the triangles (15. and 
29. i.) A p r, a d b are equiangular; and therefore 
(4. vi.) ap:pr::ad:db. Confequently, as by 
the Cor. to the Definitions ad is equal to r v, the 
reCtangle (16. vi.) under a p, d b is equal to the rect¬ 
angle under p r, r v. But d b is the diftance of the 
curve at the point d from the direCtrix ; and it is evi¬ 
dent that a p increafes as the diftance of d from the 
axis increafes. The diftance of the defcribing point d, 
therefore, from the direCtrix mud decreafe as d recedes 
from the vertex, as p r, r v are conftant; and as the 
reCtangle under a p, d b is a conftant magnitude, the 
points d and b cannot coincide. Hence the Propofition 
is evident. 

Cor, The direCtrix is alfo an afymptote to the con¬ 
choid. 


PROP. IT. 

If an ordinate he drawn from any point in either of the 
conchoids to the axis , a ftraight line drawn from the 
pole to the fame point in the curve will he a fourth pro¬ 
portional to the diftance of the ordinate from the direc¬ 
trix, the diftance of the vertex from the directrix , and 
the diftance of the ordinate from the pole. 

The reft remaining as in the preceding Propofition, 
a 2 and 


SECT. 

I. 


Fig. 1. 
2 . 
3 - 


Fig. u 
2 . 
3 * 



OF CONCHOIDS. 


SECT, and the fix firft Definitions, let d e be an ordinate to 
the axis according to the feventh Definition, and let it 
meet the axis in e 5 and then re is to r v as p e to 
p d. 

For b e is a parallelogram, and therefore (34. i.) r e 
is equal to bd, and, by the Cor. to the Definitions, 
A d is equal to r v. Confequently re: rv::bd: 
A d. But the triangles (29. i.) bda,epd are equi-> 
angular, and therefore (4. vi.) bd:ad::pe:pd. 
Confequently (n . v.) r e : r v : .* p e : p d. 

Cor. 1. The reft remaining as above, with R as a 
center and r v as a diftance defcribe v 1 c a quadrant 
of a circle, and let it cut de, or d e produced, in 1, 
and draw 1 r. Then 1 r is equal to R v, and there¬ 
fore by the above re:ir::pe:pd. In the fu- 
perior, conchoid the angle at e is common to the two 
triangles rei,ped, and in the other two conchoids, 
the angle at e in the triangle rei is equal to the an¬ 
gle at e in the triangle ped, each of them being a 
right angle. Confequently (7. vi.) the triangles rei, 
p e d are equiangular, and therefore (4. vi.) p e : e d 
: : R e : 1 e. 

Cor. 2. From the preceding Cor. the equation of 
each of the conchoids may be eafily deduced. For in 
each of them put r v = a, p r = b, r e s= x, and d e 

-y- 

Fig. 1. I. In the fuperior conchoid (47. i.) ie = v^z 2 — x\ 
as I r, r v are equal; and pe = H^. Confequent¬ 
ly b -f x :y : : x : = 1 E = — # 2 5 and j =3 

b + x x vV — ar 2 
x 

Fi S- 2 * 2. In the inferior conchoid ie = for the 

fame reafons as above 5 and p e = b «— x. Proceeding 

there- 








OF CONCHOIDS. 


therefore as in the laft article, the equation of the in- sect. 
ferior conchoid is * —— 

X 

3. In the nodatedconchoid ie = — x z } as in the Fig. 3. 

other two, and p e = b — x; and therefore the equation 

of the nodated conchoid is alfo y = . 

x 

4. The foregoing equations being cleared of the 
furd, the equation of the fuperior conchoid becomes 

+ 2 b x* + b 2 x 2 - a 2 x 2 -f y 2 x 2 — i a 1 b x = a 2 b 2 ; 
and the equation of each of the other two becomes a 1 4 
- 2 b a 3 + b l x 2 - a 2 x l +y 2 x 2 - 2 , a 2 b x = a 2 b \ 

SCHOLIUM. 

Nicomedes, the inventor of the conchoid, publiflied 
an account of an inftrument for the defcription of the 
curve *, conftru&ed upon the principles ftated in the 
firft Definition, of which the following is the fub- 
ftance. 

Let t x, p r be two rulers of wood or metal, fixed Fig. 4* 
at right angles to one another at r ; and let them be 
of indefinite length, and have each a fmooth groove to 
a convenient extent, as reprefented. Let dl be ano¬ 
ther ruler of wood or metal, of indefinite length, and 
let it alfo have a fmooth groove of a convenient extent, 
as reprefented. Let p be a pin which may be fixed in 
the ruler pr at any requifite diftance from the point 
r. Let A be a pin which may be fixed in the ruler 
dl at any requifite diftance from the extremity D. 

Let the ruler dl be adapted to the other two by 
means of the pins A, p in fuch a manner, that the pin 
a, fixed in d l, may Aide fmoothly in the groove in 
T x, and that the groove in dl may always embrace, 

* See the Oxford edition of Archimedes, page 147, 

a 3 but 










OF CONCHOIDS. 


*30 

SECT, but Hide fmoothly over the pin p, fixed in p r. Then 
L if a pencil or pen be attached to the fixed point d in 
D l, it will trace out the fuperior, inferior, or nodated 
conchoid, according as the conditions Hated in the 
fourth, fifth, or fixth Definition, are attended to in ad¬ 
juring the inftrument, 

PROP. III. PROB. I. 

T-rt'O Jlraight lines being given , let it be required to jind 
two means in continued proportion between them by a 
conchoid . 

Fig. 5. Let l g, l A be two given Hraight lines ; it is re¬ 
quired to find two means in continued proportion be¬ 
tween them. 

Let a l, l g be at right angles to one another, and 
complete the parallelogram a l g b. Bifeft a b in p, 
and b g in e. Draw l d, and, being produced, let it 
meet g b produced in h. Draw e c perpendicular to 
B g, and let it meet g c equal to ad, or d b, in c. 
Draw h c, and g f parallel to it. With c as a pole, ! 
G f as a dire&rix, and a diftance equal to A d or g c 
between the directrix and vertex, let a conchoid be - 
defcribed, and let it cut h g k in k ; and f k will be 
equal to ad or g c, by the Cor. to the Definitions,. 
Draw k l, and, being produced, let it meet b a in m. 
Then will g k, m a be the two mean proportionals re¬ 
quired. 

For (6. ii.) the rectangle under b k, k g, together ! 
with the fquare of e g, is equal to the tquare of e k ; 
and therefore, by adding the fquare of ec to thefe 
equals, the rectangle under b k, k g, together with 
the fquare of c g, (47. i.) is equal to the fquare of ck. 
By fimilar triangles m a : A b : : m l : l k : : B G : J 
G k ; and therefore (16. vi.) m a X g k is equal to ab 

X B G. 



OP CONCHOIDS. 


X b g. Again, by fimilar triangles, l g or A b : d b :: 
G h : b h, and as a b is double of d b, h g is double 
of b g, and therefore a d x h g is equal to abx bg. 
Confequently (16. vi.) ma:ad::hg:gk::cf: 
f k ; and (18. v.) m d : a d : : c K : F K. But a d 
is equal to f k, by conftru&ion, and therefore (14. v.) 
m d is equal tocK, and md 2 is equal to c k 2 . By 
the above therefore the rectangle under b k, k g, to¬ 
gether with the fquare of c g, is equal to the fquare 
of m d, which (6. ii.) is equal to the rectangle 'under 
E m, m a together with the fquare of a d, or its equal 
c g. Confequently b k x k g is equal to b m x m a, 
and (16. vi.) bm:bk::gk:ma. But by limilar 
triangles bm:bk::gl:gk::ma:al. Con¬ 
fequently (11. v.) g l : g k : : g K : M A : : m A : 
A L ; and therefore g k, m a are two means in conv 
tinued proportion between the given ftraight lines A 

E G. 


PROP. IV. PROB. II. 

An angle being given, let it be required to divide it into 
three equal parts , and by means of a conchoid . 

Let A c B be a given angle, it is required to trifeft 
it, or divide it into three equal parts. 

With c as a center, and any convenient diftance 
c A, defcribe the circle A e f, and produce the diame¬ 
ter B f indefinitely. With a as a pole, b f as a direc¬ 
trix, and a diftance between the dire&rix and vertex 
equal to ca, let a conchoid be defcribed, and let it cut 
the circumference of the circle in e. Draw a e, and 
let it meet b f in d, and the angle a d c is a third part 
of the given angle acb. 

For c e being drawn, e d c is an ifofceles triangle, 
and (5. i.) the angles edc,ecd are equal, as are 


$31 

fi 'E C T. 

I. 


Fig. 6 . 
% 



* 3 1 

SECT. 

I. 


OF CONCHOIDS. 

alfo the angles cea,cae. In Fig. 6 . the angle aec 
(32. i.) therefore is double of the angle adc, and 
confequently the angle cad is double of the angle 
adc. But the angle acb (32. i.) is equal to the an¬ 
gles cad, adc together, and therefore the angle 
A d c is a third part of the angle acb. In Fig. 7. 
the angle e a c, (32. i.) and therefore its equal aec, 
is equal to the angles edc,acd together ; and the 
angle d a c is equal to the angles a e c, a c e toge¬ 
ther. Confequently the angle d a c is equal to the 
angles edc, acd, ace taken together; that is, the 
angle d a c is equal to the angles e d c, e c d toge¬ 
ther, or to the double of the angle adc. Confe¬ 
quently, as the angle a c b (32. i.) is equal to the an¬ 
gles d a c, adc together, the angle A d c is a third 
part of the angle acb. 


J 


*' 




SECTION 





l'/utc . 1.1 / l./ntih 132 


/ 


J. Basire sc. 

















































































































J Bafire <nr • 

























































































[ 233 ] 

SECTION II. 
Of the CiffoiJ . 


LEMMA. 

Let E g F h be a circle of which c Is the center, and F‘g* 
E f, h g two diameters at right angles to one another. 

Let e f be produced to p, fo that f p may be equal to . 
c F ; and a being any point in cg draw ap, and upon 
a p as a diameter defcribe the femicircle abp. Then 
if from a a ftraight line a b, equal to c p, be infcribed 
in the femicircle, the point in which a b is bife&ed 
will be on the fame fide of c p with the point A. 

For let k be the point in which A p is bife£ted, and 
draw k f ; and draw k d perpendicular to a b. Then 
as a c p is a right angle, the circumference abp 
(31. iii.) pafles through c ; and as A b, c p are equal, 
the circumference acb (28. iii.) is equal to the cir¬ 
cumference pbc; and therefore the circumference A c 
is equal to the circumference b p. The angle apc 
(27. iii.) is therefore equal to the angle pab; that is, 
the angle kpf is equal to the angle k a d. Again, 
as p k is equal to k a, and p f equal to f c, p k : k a 
: : p f : f c ; and therefore (2. vi.) k f is parallel to 
c a, and (29. i.) the angle p f k is a right angle. The 
triangles p k f, a k d are therefore equiangular, and 
(26. i.) k f is equal to K d, and A D to p f. Confe- 
quently if with k as a center and k f as a diftance a 
circle be defcribed, it will pafs through d, and c P will 
be a tangent to this circle (16. iii.) as it is at right an¬ 
gles to k f. The ftraight line A b is alfo bife&ed in d. 



234 OF THE CISSOID. 

SECT, for A d is equal to p f. Confequently the point in 
iL which A b is bife&ed is on the fame fide of c p with 
the point A. 


DEFINITIONS. 

I. 

Fig- 9- Let e G f h be a circle, of which c is the center, and 
E F, H G two diameters at right angles to one another. 
Let e f be produced to p, fo that f p may be equal to 
c f the radius, and let h g be produced indefinitely on 
one fide towards a a point in h g. Let the ftraight 
line b l, of indefinite length towards l, pafs through 
p; and let b a, equal to e f or c p, be at right angles 
to e l, and let d be the point in which b a is bifeeted. 
Then if the ftraight lines b l, b a be fo moved that 
e l always pafs through p, and the extremity a of b A 
be always in h g, the point d will defcribe a curve 
F g d which is called a Ciffoid) or the Ciffoid of Dio~ 
$les* 

II. 

The circle e g f h is called the Generating Circle of 
the ciffoid 3 and the point p is called its Pole . 

III. 

The ftraight line h g a is called the DireSlrix of the 
ciffoid. 


IV. 

The diameter f. f of the generating circle is called 
the Axis of the ciffoid, and the point f is called its 
Cuff. 

V. 

The ftraight line e k perpendicular to the axis is 
called the Afymftote to the ciffoid. 

VI. 

A ftraight line drawn from any point in the ciffoid 
perpendicular to the axis is called an Ordinate to the 


axis, 




OF THE CISSOID. 


axis, and the fegment of the axis between the cufp and SECT, 
an ordinate is called an Abfcifs of the axis. IL 

PROP. I. 

'The cijjoid commences at the cufp , and the curve is entirely 
on one Jide of the axis • it alfo paffes through the point 
in which the direbirix cuts the generating circle , and 
being continually produced it approaches nearer and 
nearer to the ajymptole , hut never meets it. 

For, the reft remaining as in the Definitions, draw Fig. % 
the ftraight line p a. Then as p c a, pea are right 
angles, a femicircle defcribed upon pa as a.diameter 
will (31. iii.) pafs through b and c; and the ftraight 
line A e, equal to c p, will be infcribed in this femi¬ 
circle, in every fituation of a, regulated according to the 
fir ft Definition. Confequently when a coincides with 
c the ftraight line a b will coincide with c p, and the 
point d will coincide with f. If therefore the defcrip- 
tion of the curve be fuppofed to begin from this fitua¬ 
tion, the cufp f will be the point at which it com¬ 
mences, and as foon as a has moved from c towards G 
the defcribing point d will be removed from e f to¬ 
wards G, according to the Lemma prefixed to the De¬ 
finitions. When the diftance of a, in the diredtrix, 
from c is equal to c p or a b, then the point b will co¬ 
incide with c, and the defcribing point d will coincide 
with g. Through d draw k m perpendicular to c a, 
and let it meet the afymptote in k. Let b p cut c a 
in n. Then as p c n, n b a are right angles, and 
(15. i.) as the angles p n c, a n b are equal, the trian¬ 
gles p c n, A b n are equiangular. The triangles pcn, 
amd are therefore equiangular, and the angle cpn 
is equal to the angle m a d. But as the point a re¬ 
cedes from g the point n alfo recedes from it, and 

there- 



236 of the cissoid. 

SECT, therefore the ciffoid being continually produced the 

1I# angle p n c becomes lefs and lefs, and confequently 

the angle c p n, or its equal m a d, becomes greater 
and greater. The perpendicular d m mud therefore 
continually increafe, as the length of d a is conftant, 
and confequently d k, the diftance of the curve from 
the afymptote, mud continually decreafe, as km (34.i.) 
is equal to ec. The point d however can never fall 

into e k, as d m can never become equal to da or 

EC) for if it did then p b would become parallel to 
c A ) which is impoffible. 

Cor . P'rom the above it is evident, that an ordinate 
drawn from any point in the ciffoid to the axis will cut 
the generating circle. 

PROP. II. 

An ordinate , drawn from any 'point in the ciffoid to the 
axis, is a third proportional to its fegment between the 
generating circle and the axis , and the correfponding 
ahfcifs . 

Frg. 10. From d, any point in the ciffoid f d g whofe cufp is 
ll ° f, let d r be drawn an ordinate to the axis f e, and 
let it cut the generating circle" in t ; then t r is to 
R F as R F to R D. 

v For, the reft remaining as in the Definitions and pre¬ 
ceding Propofition, let d r cut the generating circle 
again in s, and draw c s. Then as A d is equal to c s, 
and (34. i.) d m equal to r c, and d m a, c r s right 
angles, the fquares of m a, s r are (47. i.) equal ; and 
confequently m a, s r are equal. Again, as in the 
triangles abn, pcn, ab,fc are equal, and as the 
angle b n a is equal to the angle c n p, and the angle 
abn equal to the angle pcn, the fide b n is equal to 
the fide c n. v By fimilar triangles alfio d a : m a : : 

N A 



OP THE CISSOID. 


2 37 


n A : b A, and md:ma::nb:ba. Confequently sect. 
on account of the equals (24. and its fir ft Cor. v.) f r : IL 

s R : : c a : E F ; and f r x e f is equal to s r x c a. 

But (3. ii.) f r X e f is equal to er x r f together 
with the fquare of f r ; and (35. iii.) e r x r f is 
equal to the fquare of s r. Confequently f r x e f is 
equal to the fum of the fquares of f r, s r. Again, 
sr X c a is equal to s R x s d, for (34. i.) r d, c m 
are equal, and by the above m a is equal to s r. But 
s r x s d (3. iii.) is equal to s r X r d together with 
the fquare of s r. By the above therefore the fum of 
the fquares of f r, s r is equal tosRXRD together 
with the fquare of s r. Confequently s r x r d is 
equal to the fquare of fr; and therefore (3. iii.) as 
T r is equal to s r, (17. vi.) tr;rf::rf:rd« 

Cor . 1. The ftraight lines e r, r t, r f, r d are in 
continued proportion. For (Cor. 8 . vi.) erjrt:: 
r t : rf; and as above r t ; r f : : R f : r d. 

Cor . 2. The equation of the ciffoid is eafily deduced 
from the laft Corollary. For put ef — a, f r = x, 
and R D = y \ and then e r = a — x, and r t 2 = e r 
X R F, or r T = \fax-x 1 . Confequently *Jax—x z 

: x : : x :— - ■•■■■ = y, and- 


~—y , or-= y 

" a — x y 


\/ a x — x* 

and therefor .v 3 = ay* — xjy z 3 which is the equation of 
the curve. 


PROP. IIL 

If from the cufp of a ciffoid a Jlraight Vine he drawn cut- 
t'uig the ciffoid and the generating circle, Jlraight lines 
drawn through the points ofJ'eEtion, and perpendicular 
to the axis , will he equally dijlant from the direBrix . 

Let f d g l be a ciffoid, of which f is the cufp, f e Fig. 12. 
the axis, h c g the diredtrix, and e g f h the generat¬ 
ing 








S3 8 

SECT. 

II. 


OF THE CISSOID* 

ing circle. From f draw any flraight line f k cutting 
the cifloid in d and the circle in k, and let d b, k m 
be perpendicular to the axis; the flraight lines d b, 
k m are equally diftant from h c g. 

For let c be the center of the circle, and let the di¬ 
rectrix cut the circle and cifloid in g. Let b d meet 
the circle in a, and let k m meet it again in n, and 
draw e k. Then, by Prop. II. ae:bf::bf:bd; 
and (4. vi.) b f : b d : : m f : m k. Confequently 
(11. v.) A b : B f : : m f : m k ; and therefore ( 6 . vi.) 
the triangles a f b, f k m are equiangular, and the an¬ 
gle a f b is equal to the angle m kf. The circumfe¬ 
rence A e therefore (26. iii.) is equal to the circumfe¬ 
rence n f, which is equal to the circumference f k. 
The circumference f a is therefore equal to the cir¬ 
cumference e K, and confequently (29. iii.) the flraight 
lines f A, e k are equal. But (Cor. 8. vi.) e m is a 
third proportional to e f, e k ; and f b is a third pro¬ 
portional to e f, a f. Confequently e m, f b are 
equal, and therefore c m is equal to c b. 

Cor. 1. Hence it is evident that the arch gk is equal 
to the arch G a . 

Cor. 2. If equal arches as G ic, g a be fet off in the 
circumference of the circle, on theoppoflte fldes of h g, 
and perpendiculars k m, a b be drawn to the diameter 
e f, a flraight line drawn from f to k will cut the per¬ 
pendicular a b and the cifloid in the fame point. For 
the fame reafons a flraight line drawn from f to A will 
cut the perpendicular m k and the cifloid in the fame 
point. For it may be proved, as above, if the flraight 
line f a cut the generating circle in a and the cifloid 
in l, and if a b, l m be drawn perpendicular to the 
axis, that c e, c m are equal. 


vSCIIO- 



OP THE CISSOID. 


2 39 


SCHOLIUM. sect. 

11. 

Diodes, the inventor, confidered the property ex- 

prefled in the laft Corollary as the primary one of the 
cifloid; and he fuppofed the defcription of the curve 
to be effected by means of an indefinite number of 
points, as d and l, obtained from equal arches as g k, 
g a, and the interle&ions of the flraight lines fk,fa 
with the perpendiculars a b, m l *. Sir I. Newton 
firft (hewed how the cifloid might be defcribed by con¬ 
tinued motion according to the conditions exprefled in 
the firft Definition in this fe&ionf. The method will 
be eafily underftood if p c a, l b a be fuppofed to be Fig. 9. 
two fquares of wood or metal, p being a fixed point in 
p c one arm of the one, and A a fixed point in b a an 
arm of the other; and if the adjuftment of the inftru- 
ment and its action be fuppofed to be regulated as 
mentioned in the firft Definition. 

It is evident from the Definitions, and the Propofi- 
tions and their Corollaries, that with the fame gene¬ 
rating circle egfh, the fame pole p, and the fame 
cufp f, another cifloid fur may be defcribed on the 
oppofite fide of ep to that on which fg d is defcribed. 

It is alfo evident that s e k, touching the generating 
circle in e, is the common afymptote to thefe two cil- 
foids, and that e f is their common axis ; and it may 
readily be perceived that the properties proved of the 
one equally apply to the other. 

* See the Oxford edition of Archimedes, page 138. 

f See the Appendix to the Arithmetica Univerfaljs. 


SECTION 






SECTION III. 

Of the Quadratrix. 


f ;&. 13. 


% 


DEFINITIONS. 

I. 

Let A d e be a femicircle* and let c be the center of . 
the circle* and a e a diameter. Let c d be at right 
angles to A e and meet the, circumference in d, and 
let it be produced to s fo that d s may be equal to.CD, 
Let a ftraight line c h of indefinite length revolve 
about c, begin its revolution from a coincidence with 
ce, and move towards d ; and at the fame time that 
c h begins to revolve let a ftraight line f g move from 
a coincidence with € e towards s* and let fg be al¬ 
ways parallel to A e* or perpendicular to c s. Let ch 
revolve and f g move with an uniform velocity* and 
let the arch e h pafled over by the revolution of ch 
be to the diftance c f moved over by the extremity" of 
f g in the fame time* as the quadrantal arch ed to the 
radius c e 3 the curve bid defcribed by the point 1* 
in which c h* f g cut one another* is called a Quadra - 
trixy or the Quadratrix of Dinofir a tes. 

II. 


The circle a d e is called the Generating Circle of 
the quadratrix. 

III. 

The ftraight line c s is called the Axis of the qua¬ 
dratrix. 


IV. 


If b be the point at which the lines ch,fg com¬ 


mence 





OF THE aUADRATRlX. 


mence their interfe&ion, the ftraight line c B is called SECT, 
the Bafe of the quadratrix. I1L 

V. 

The flraight line s t perpendicular to the axis is 
called the AJymptote to the quadratrix. 

PROP. L 

The curve of the quadratrix pajfes through the point in 
which the axis cuts the generating circle ; and being 
continually produced it approaches nearer and nearer to 
the ajymptote, but never meets it. 

Part I. Every thing remaining as in the Definitions, Fig- * 3 * 
as c H revolves and f g moves with an uniform velo¬ 
city, and as the velocity with which c h revolves is to 
the velocity with which f g moves as the quadrantal 
arch e d to the radius c £>, the arch eh is to c f as 
the quadrantal arch ed to the radius c d. When the 
revolving line c h therefore coincides with c d, the 
point F in f G will coincide with d, and i will alfq 
coincide with d. Confequently the curve of the qua¬ 
dratrix muft pafs through d. 

Part II. The reft remaining as above, let c l repre- 
fent the revolving line after it has proceeded beyond 
c d from e, and let l m reprefent the fttuation of the 
line moving parallel to A e at the fame time, and let 
c l, m l interfe£t one another in l, and let C l cut the 
generating circle in k. Then it is evident, from the 
firft Definition, that the point l is in the curve of the 
quadratrix ; and, for the fame reafons as above, the 
arch e d k is to c m as the quadrantal arch e d to the 
radius c d. Hence it is evident, that, if the curve of 
the quadratrix be continually produced, the revolving 
line will cut off greater and greater arches of the ge¬ 
nerating circle, reckoning from the extremity e, and 
r con-* 



OP THE QUADRATRIX. 


SECT. 

III. 


Fig. 14. 


confequently the diftances of the moving line parallel 
to A e muft become greater and greater. The diftance 
of the moving line parallel to a e from the afymptote 
T s muft therefore continually decreafe. Confequently 
the curve of the quadratrix muft approach nearer and 
nearer to the afymptote upon being continually pro¬ 
duced, but they can never meet, for if they did then 
the revolving line would coincide with a e, and A e, 
t s would meet. But this is impoffible, for, by the 
fifth and firft Definitions, (and 28. i.) they are parallel. 

Cor . Any arch e h and diftance c f, paffed over by 
the revolving line c h and moving line f g in the fame 
time, are to one another as the quadrantal arch e d to 
the radius c d. Alfo (19. v.) the arch d h is to d f as 
the quadrantal arch ed to the radius cd. 

PROP. II. 

The bafe of the quadratrix is a third proportional to ths 
quadrantal arch of the generating circle and its radius . 

Let b 1 d be a quadratrix, of which c e is the bafe* 
A d e the generating circle, and c e or cd the radius 
of the circle, c being its center; the quadrantal arch 
E h d of the generating circle is to its radius c e as 
c e to c B. 

For let c h be a pofition of the revolving line, and 
f 1 a correfponding pofition of the line which moves 
parallel to a e, and let them interfeft one another in i, 
as in the firft Definition, fo that 1 may be in the curve 
of the quadratrix. Let c h, f t be indefinitely near to 
c e, fo that the arch e h of the generating circle may 
be indefinitely fmall5 and let he, i k be perpendicu¬ 
lar to c E. Then, by the Cor. to Prop. I. the arch e h : 
c f : : arch e H d : c E. But it is evident that f k is 
a parallelogram, and therefore (34. i.) c f is equal to 



OP THE QUADRATE.IX* 


H3 


K 15 and as the arch eh is indefinitely fmall, it is SECT, 
equal to its fine l h. Confequently l h : K I : : arch IIL 

E H d : ce ; and therefore (n. v. and 4. vi.) arch 
ehd:ce::cl:ck. But when the arch eh is in¬ 
definitely fmall, and equal to l h, c l is equal to the 
radius, and c k becomes equal to c b. Confequently 
the quadrantal arch ed:ce::ce:cb. 

Cor. 1. As by the above and inverfion cb:ce:: 
c e : the quadrantal arch e d, by the Cor. to Prop. I. 

(and 11. v.) ce : cd : : fd : the arch d h. Alfo by 
the above, Cor. to Prop. I. (and 11. v.) c b : c d : : 
c f : the arch e h . 

Cor. 1. The equation of the quadratrix may be ob¬ 
tained from the above in the followi g manner. Put 
c d — a, c b — b 9 e h = z, c f = y; and then, as 

a 2 

in the Propofition b : a : : a : —g — the quadrantal 


arch e d . Confequently ~ : a : 

O %.* 

and b z = ay , the equation of the curve 


oz 

z : — = c F = y ; 
a 


LEMMA. 

A circle is equal to a right angled triangle, which 
has one of the fides round the right angle equal to the 
radius of the circle, and the other fide equal to the cir¬ 
cumference. 

Let a 1 b d e be a circle, as in Fig. 15. of which c 
is the center, and c 1 a radius, and let k m n, as in 
Fig. 16. be a right angled triangle, having k m one of 
the fides round the right angle at m equal to c 1, and 
the other m n equal to the circumference of the circle; 
the circle A 1 b d e is equal to the triangle k m n. 

For, if it be poffible, let the circle be greater than 
the triangle, and firft by infcribing a fquare in it, and 
afterwards by a repeated bife&ion of circular arches, 
R % let 





244 

SECT. 

III. 


OP TtlE QUADRATRIX. 

let a polygon of an even number of Tides be infcribed 
in the circle ; and let the excefs of the circle above 
the polygon be lefs than its excels above the triangle. 
Then the polygon thus infcribed in the circle will be 
greater than the triangle. Let i b be a fide of this po¬ 
lygon, and let c l, at right angles to it, meet it in l. 
Then c l is lefs than k m, and as the (Iraight line i b 
is lefs than the circular arch i b, the perimeter of the 
polygon is lefs than the circumference of the circle. 
The rectangle under c l and the perimeter of the po¬ 
lygon is therefore lefs than the rectangle under k m, 
m n. But it is evident (from i. ii. and 34. i.) that the 
reftangle under c l and the perimeter of the polygon 
is double the area of the polygon ; and the rectangle 
under k m, m n is double the area of the triangle 
k m n. The polygon is therefore lefs than the trian¬ 
gle K m n ; and it is alfo greater ; which is abfurd. 

But, if it be poffible, let the circle be lefs than the 
triangle; and firft by defcribing a fquare about the 
circle, and afterwards by a repeated bifection of circu¬ 
lar arches let a polygon be defcribed about the circle, 
and let the excefs of the polygon above the circle be 
lefs than the excefs of the triangle above the circle. 
And that this may be done is evident from (1. x. and 
1. xii.) confidering that if b h, a h touch the circle in b 
and a and meet one another in h, then if c h be drawn 
cutting the circle in 1, and f g touch it in 1 and meet 
b h in f and a h in g, and b i, i a be drawn, the tri¬ 
angle h 1 f is greater than the triangle bif, and the 
triangle h i g is greater than the triangle aig. For 
(18. iii.) h 1 f, h 1 g are right angles, and therefore 
(18. i.) f h is greater than f 1, and g h greater than 
g 1. But it is evident (from 36. iii.) that b f is equal 
to f 1, and g 1 to g A, and therefore (1. vi.) the trian¬ 
gle h 1 f is greater than the triangle bif, and the 

tri- 



OF THE aUADRATRIX. 

triangle h i g is greater than the triangle A i G. Let 
f g be a fide of the polygon defcribed about the circle, 
whofe excefs above the circle is lefs than the excels of 
the triangle k m n above the circle, and then the poly¬ 
gon, of which f G is a fide, is lefs than the triangle 
k m n. But as the perimeter of the circumfcribed po¬ 
lygon is greater than the circumference of the circle, 
and as the rectangle under c i and the perimeter of the 
circumfcribed polygon is equal to the double of the 
area of the polygon, it is evident for the fame reafons 
as above that the circumfcribed polygon is greater 
than the triangle k m n. The circumfcribed polygon 
therefore is both lefs and greater than the triangle 
K m n 5 which is abfurd. Confequently the circle 
A i b d e is equal to the triangle k m n. 

Cor. i. A circle is equal to a re&angle which has 
one of its fides equal to the radius of the circle, and 
the other fide round the fame angle equal to half the 
circumference. 

Cor. 2. The diameter of one circle is to the diame¬ 
ter of another as the circumference of the firfl: men¬ 
tioned to the circumference of the other. For put d 
equal to the diameter of the one and c equal to its cir¬ 
cumference, and put d equal to the diameter of the 
other and c equal to its circumference. Then, by the 
preceding Cor. (and 2. xii.) d x c : d x c : : d* : d z 9 
and d x c : D z : ; d x c : d z . Confequently (1. vi.) 
c : d : : c : d. 

Cor. 3. If the two circles fhge, klme have the 
common center c, and c a, c b be drawn cutting the 
outer circle in A, b, and the inner in D, e, the radius 
c b is to the radius c e as the arch ab to the arch d e. 
For let f G, h b be two diameters of the outer circle 
at right angles to one another, and let f g cut the in¬ 
ner circle in k, aj, and h b cut it in l, e. Then 
R 3 (33< 


24 5 


SECT. 

HI. 


Fig. xr» 



2\S 


OF THE QUADRATRIX. 


SECT. 

III. 


Fig. 19. 


(33 • vi.) the arc ^ F B : the ar °h A B : : the an g^ e fcb: 
the angle ace : : the arch k e : the arch d e ; and 
therefore, by alternation, the arch f b : the arch k e : : 
the arch a b : the arch d e. But the arch f b is a 
fourth part of the circumference f h g b, and the arch 
k e is a fourth part of the circumference k l m e. 
Confequently, by the lad Cor. (and 15. v.) cb : c E : : 
the arch a b : the arch d e. 

Such fe&ors as a c b, d c e which have the angles 
at their centers equal, are called Similar Seniors. 

PROP. III. 

If Jrom any point in the quadratrix a Jlraight line he 
drawn to the center of the generating circle , and alfo a 
Jlraight line perpendicular to the axis , and if with the 
center of the generating circle , and the bafe as a dif- 
tance , a circle he defcrihed , tie arch of this circle inter¬ 
cepted between the extremity of the bafe and the Jlraight 
line drawn from the quadratrix to the center , will be 
equal to the fegment of the axis between the center and 
perpendicular . 

From any point l in the quadratrix b d l let a 
draight line l c be drawn to c the center of the gene¬ 
rating circle a d e, and with c as a center and cb, the 
bafe of the quadratrix as a didance, let a circle be de- 
fcribed; the arch bf of this circle, intercepted be¬ 
tween b and c l, is equal to c m, the fegment of the 
axis com between c the center and l m the perpen¬ 
dicular. 

For, by Prop. II. and inverfion, c b : c d : : c D : 
the quadrantal arch n e; and, by Cor. 1. to Prop. II. 
c d : quadrantal arch d e : : c m : the arch e k ; and, 
by Cor. 2. to the preceding Lemma, (and 15. v.) cb : 
c d or c e i : the quadrantal arch b g : the quadrantal 

arch 



OF THE QUADRATRIX. 


547 


arch de. Again (33. vi.) the angle ecd: the angle sect, 
e c k : : the quadrantal arch e d : the arch e k ; and ’ 
for the fame reafons the quadrantal arch b g is to the 
arch b f in the fame proportion. Confequently (11. v.) 
the quadrantal arch e d : the arch e k : : the qua¬ 
drantal arch e G : the arch b f ; and, by alternation, 
the quadrantal arch e d : the quadrantal arch eg:: 
the arch e k : the arch b f. By the above therefore 
(and ir. v.) the arch b f : the arch e k : : c m : the 
arch e k ; and (14. v.) confequently the arch b f is 
equal to c m. 

Cor. The quadrantal arch b g is equal to the radius 
c d. For, as above, by Prop. II. cb:cd::cd: 
the quadrantal arch d e ; and, by Cor. 2 to the pre¬ 
ceding Lemma, c b : c d : : the quadrantal arch b g : 
the quadrantal arch e d. Confequently (11. v.) c d : 
the quadrantal arch e d : : the quadrantal arch b g : 
the quadrantal arch e d ; and therefore (14. v.) c D is 
equal to the quadrantal arch b g. 

SCHOLIUM. 

If the ftraight line c h revolve, and the ftraight line Fig. 18, 
f g move with the fame relative velocities as ftated in 
the firft Definition, but on the fide of a e oppofite to 
that before fuppofed, the quadratrix l d b may be ex¬ 
tended as represented by l d b k o in Fig. 18. and if 
the generating circle be completed, and dc be pro¬ 
duced to m fo that c M be equal to c s, then a ftraight 
line m n at right angles to c m will alfo be an afymp- 
tote to the curve. It is alfo evident that the curve 
will pafs through the point K in which c m cuts the 
generating circle. 

It is much to be wiftied that an inftrument were de- 
vifed for defcribing the quadratrix by continued mo¬ 
tion, as the two following important Problems could 
r 4 then 



24S 

SECT. 

III. 


Fi S* *3* 


Fig. 18, 


OP THE QUADRATRIX. 

then be folved geometrically. In the following folu- 
, tions the poffibility of defcribing the quadratrix is ne- 
cefTarily fuppofed. 

PROP. IV. PR OB. I. 

From a given reBilineal angle to cut off' any fart required , 
by means of the quadratrix . 

Let k c e be a given re&ilineal angle; it is required 
to cut off any part from it, by means of the quadratrix. 

With c as a center, and any convenient diftance c e, 
let a circle ade be defcribed. With the generating 
circle ade let the quadratrix b i d l be defcribed, 
and let it meet c k in l. Let cdm be the axis of the 
quadratrix, and let l m be perpendicular to it. Let 
c m be fo cut in f (9. vi.) that c m may be to c f as 
the whole given angle k c e to the part required. 
Draw f g parallel to c e, and let it cut the quadratrix 
in 1. Draw c 1 and let it meet the circle in h ; and 
the angle e c h will be the part required. 

For, by the Cor. to Prop. I. (and 11. v.J’cm : cf: : 
the arch k e : the arch h e ; and (33. vi.) the arch 
k e : the arch he: : the angle kce : the angle h c e. 

PROP. V. PROB. II. 

To find a fquare equal to a given circle , by means of the 
quadratrix . 

Let a d e k be a given circle ; it is required to find 
a fquare equal to it, by means of the quadratrix. 

Let c be the center of the circle, and with a d e k 
as a generating circle, and its diameter d k as an 
axis, let the quadratrix l d b k o be defcribed, 
and let c b, in the diameter ace, be its bafe. 
Through b draw p a parallel to d k, and let it meet 

D P 



OF THE QUADRATRIX. 


H9 


D P perpendicular to d k in p, and k a perpendicular 
to d k in a. Draw c p, c q, and let them meet rev 
parallel to d k in r and v. Draw r w, v x perpendi¬ 
cular to d k. Then the right angled parallelogram 
w R v x is equal to the given circle a d e k, and a 
mean proportional between the fides w r, R v is the 
fide of the fquare required. 

For with c as a center, and c b as a diftance, let the 
circle g b i be defcribed, and let it meet d k in g and 
i. Then, by the Cor. to Prop. III. the quadrantal 
arch b g is equal to c d, and the quadrantal arch b i is 
equal to c k. Again, by the above conftru£tion, d b, 
w e are parallelograms, and therefore (34. i.) d p is 
equal to c b, and w r is equal to c e. The triangles 
c d p, c w R are alfo equiangular, (29. i.) and there¬ 
fore (4. vi.) D p : w R : : c D : c w. Confequently, 
by Cor. 2. to the Lemma in this fe£tion, (and 15. v.) 
the quadrantal arch d e is equal to c w, and for the 
fame reafons the quadrantal arch k e is equal to c x ; 
and therefore w x is equal to half the circumference of 
the circle adek. The right angled parallelogram 
w R v x is therefore, by Cor. 1. to the Lemma in this 
fe&ion, equal to the circle adek; and a mean pro¬ 
portional (13. vi.) being found, it will be equal to the 
fide of the fquare required. 

Cor . It is evident, from the Cor. to Prop. III. and 
Cor. 1. to the Lemma in this Se6tion, that the right 
angled parallelogram d p q k is equal to the circle 
GUI. 


section 



SECT. 

III. 



SECTION IV. 

Of Cycloids . 


DEFINITIONS. 

I. 

Tig. so. If the circle Awe roll along the ftraight line A b fo 
that every part of the circumference may touch it in 
regular fuccefiion, and if at the commencement of the 
' revolution of the circle it touch A b in a, and if du¬ 
ring the revolution the point A remain fixed in the cir¬ 
cumference, and at the end of it meet the ftraight line 
in b m } the curve avb defcribed by the point A, during 
the revolution, is called a Cycloid . 

II. 

The circle A w e is called the Generating Circle of 
the cycloid. 

III. 

The ftraight line A b is called the Bafe of the cy¬ 
cloid. 

Cor. As every part of the circumference of the cir¬ 
cle touches the bafe, in regular fucceflion, the bafe of 
the cycloid is equal to the circumference of the gene¬ 
rating circle. 

IV. 

The ftraight line h v, bifefting the bafe A b at right 
angles and meeting the curve in v is called the Axis of 
the cycloid. 

V. 

The point v is called the Vertex of the cycloid. 

VI. 

A ftraight line drawn from any point in the curve 

per- 






J.Basirc sc. 


































































































OP CYCLOIDS. 


25 * 


perdendicular to the axis is called an Ordinate to the SECT, 
axis 5 and the fegment of the axis between the vertex IV ‘ 
and an ordinate is called an AbJ'cifs . 

PROP. I. 

During the revolution of the generating circle its center 
defcribes a Jlraight line equal to the bafe of the cycloid ; 
and the axis of the cycloid is equal to the diameter of 
the generating circle 

For, the reft being as in the Definitions, let c be the 
center of the generating circle, and a e the diameter 
paffing through a. Then as the circle awe always 
touches the ftraight line a b, the center c is at the 
fame diftance from a b throughout the revolution, and 
therefore it muft defcribe a ftraight line. At the com¬ 
mencement of the revolution, and alfo at the end of it, 

Ac (18. iii.) is perpendicular to a b, and therefore it 
is evident that if thefe perpendiculars and the line de¬ 
fcribed by the center were drawn, a parallelogram 
would be formed, of which the line defcribed by the 
center and a b would be oppofite fides. The ftraight 
line defcribed by the center is therefore equal to the 
bafe of the cycloid. Laftly, let a w e be the pofition 
of the generating circle at the beginning of the revo¬ 
lution, and then it is evident, from the firft five Defini¬ 
tions, that at the middle of the revolution the point e 
will coincide with h and the defcribing point A with 
v. The axis H v therefore of the cycloid is equal to 
the diameter of the generating circle. 

Cor . A circle defcribed on the axis of a cycloid is 
equal to the generating circle. 


PROP. 



OF CYCLOIDS. 


353 - 

sect. 

IV. 


Fig. 20. 


PROP. II. 

_ An ordinate drawn from any point in the cycloid to the 
axis is equal to the arch of the generating circle , de- 
fcribed on the axis, between the vertex and ordinate , to¬ 
gether with the fine of the fame arch . 

From any point f in the cycloid avfb let f m be 
drawn an ordinate to h v the axis* and let it cut the 
circle vph defcribed on the axis in the point p ; the 
ordinate f m is equal to the arch v p, between v the 
vertex and the ordinate, together with p ivkthe fine of 
the fame arch. 

For let s f l i represent the generating circle when 
the defcribing point coincides with f, and in this 
fituation let t denote its center, and let l be the 
point in which it touches a b. Draw the diameters 
ltSj f t i, and let l t s meet f m in n. Then s l 
(18. iii.) is perpendicular to ab; and, as f m is per¬ 
pendicular to v h, m l is a parallelogram, and therefore 
(34. i.) M h is equal to n l, and m n is equal to h l. 
Rut, from the Cor. to the third Definition, the femi- 
circle 1 l f is equal to h b half the bafe, and from 
the defeription of the curve the circular arch lf is 
equal to the remaining part lb of the bafe. Confe- 
quently h l is equal to the arch 1 l, or to its equal 
(26. iii.) s f ; and as h m, l n are equal, it is evident 
(from Cor. 8. vi.), that m p is equal to n f, and 
the arch v p equal to the arch s f. The arch v p is 
therefore equal to m n ; and as n f is equal to p m, the 
ordinate f m is equal to the arch v p together with 
p m the fine of the fame arch. 

Cor. The equation of the cycloid is immediately 
obtained from the above. For put f m = y, the arch 
vp = «, and p m = s , and then z + s =y, the equa¬ 
tion of the curve. 

DE- 



OF CYCLOIDS. 


DEFINITIONS. 

VII. 

Let a w e be a circle in which c is the center, a e 
a fixed diameter, and d a point in a e or in a e pro¬ 
duced ; and let the circle awe roll along the flraight 
line ab fo that every part of the circumference may 
touch it in regular fucceflion from the beginning of 
the revolution at a to the end of it at e ; the curve 
d v f g deferibed by the point d during the revolution 
is called a Curtate Cycloid if the point d be without 
the circle as in Fig. si. but if d be within the circle 
as in Fig. 24 . the curve d v f g is called a Prolate or 
Tnfieftcd Cycloid . 

VIII. 

The circle A w e is called the Generating Circle ei¬ 
ther of the Curtate or Prolate Cycloid. 

IX. 

The flraight line d g joining the points in which 
the generation of the curve begins and ends is called 
the Bafe of either of the two Cycloids. 

X. 

The flraight line r v bife61ing the bafe d g at right 
angles and meeting the curve in v is called the Axis, 
and the point v is called the Vertex of either of the 
two cycloids. 

XI. 

A flraight line drawn from any point in the curve 
perpendicular to the axis is called an Ordinate to the 
axis; and the fegment of the axis between the vertex 
and an ordinate is called an Abfcifs. 

PROP. III. 

In either the curtate or prolate cycloid the bafe is equal to 

the. circumference of the generating circle \ and in the 

curtate 


* 5 * 


SEC T. 
IV. 


Fig. 23 m 

2 4 -' 



<n 


254 


OP CYCLOIDS. 


T. 


Fig. 21. 
24. 


curtate cycloid the axis is greater , hut in the prolate it is 
lefs , than the diameter of the generating circle . 

For the reft remaining as in the Definitions, the ge¬ 
nerating circle both at the beginning and end of the 
revolution touches the flraight line a b ; and therefore 
dab (18. iii.) is a right angle. And, as at the end of 
the revolution ad is reprefented by bg,bg is equal 
to a d, and a B G is a right angle. Confequently 
(28. i.) a d, b g are parallel, and therefore (33. i) 
A b, d g are equal and parallel. But as the revolution 
of the generating circle begins at a and ends at b, the 
circumference of the circle is equal to a b, and there¬ 
fore the bafe d g is equal to the circumference of the 
generating circle. Again, as r y bife&s d g at right 
angles, n h is parallel to d a or g b, and therefore a b 
is bife&ed in h, and a h is equal to half the circumfe¬ 
rence of the generating circle. Confequently in the 
middle of the revolution the point e will coincide 
with h, and the diameter e a as to pofition will coin¬ 
cide with the axis r v. Hence it is evident, that in 
the curtate cycloid the axis r v is greater than the dia¬ 
meter of the generating circle by the double of ad; 
but in the prolate cycloid the axis r v is lefs than the 
diameter of the generating circle by the double of 

A D. 

Cor. From the above it is evident that in the middle 
of the revolution the center of the generating circle 
bife&s the axis r y ; and it is alfo evident that during 
the revolution the center of the generating circle is at 
the fame diftance from the bafe d g. 

PROP. IV. 

If an ordinate be drawn from any point in the curtate or 



OP CYCLOIDS. 


prolate cycloid to the axis arid cut a circle defcribed on SF.c t. 
the axis as a diameter , the arch of the circle between IV ‘ 
the vertex and ordinate will be to the fegment of the or- 
dinate between the cycloid and circle as the circumference 
of the circle tv the lafe of the cycloid . 

Let d v g be a curtate or prolate cycloid, of which Fig. 21. 
D o is the bafe, r v the axis, and v the vertex, and 
from any point f in the cycloid let f m be drawn an 
ordinate to the axis, and let it cut the circle v p r, de¬ 
fcribed upon v R as a diameter, in p ; the arch v p is 
to the fegment f p as the circumference of the circle 
V p R to the bafe d g. 

For, the reft remaining as in the Definitions, let T 
be the center of the generating circle when the de- 
feribing point D coincides with f. Through t draw 
the ftraight line s l a perpendicular to d g, and let it 
meet a b in l. Then, as a b, d g are parallel, x l is 
perpendicular to a b ; and, by the Cor. to the preced¬ 
ing Prop, and the defeription of the curve, x l is equal 
to the radius of the generating circle, and t a is equal 
to the radius of the circle vpr, Through the points 
F and t draw the ftraight line fki, With t as a 
center and t q, as a radius let the circle s f a k be de¬ 
fcribed ; and with the fame center and t l as a radius 
let the arch l i be defcribed. Then, for the fame rea- 
fons as in the demonftration of the fecond Propofition, 
m l is a parallelogram, pf is equal to h l, and the 
arch v p is equal to the arch s f. But the arch s F 
{2,6. iii.) is equal to the arch a k ; and as x l is equal 
to c a, from the generation of either cycloid, the arch 
l 1 is equal to the ftraight line h l. Again, as x q k, 
t l 1 are fimilar fe&ors, by Cor. 2. and 3. to the Lem¬ 
ma in the preceding Se&ion, (and 11. v.) the arch k a 
; the arch l 1 : : circumference of the circle sfqk: 

the 



OF YCLOIDS. 


2 S 6 


S E c T. the circumference of the circle l i. Confequently, on 
account of the equals, the arch y p : the fegment f p 
: : the circumference of the circle v p r : the bafe dg. 

Cor . The equation of the curtate cycloid, and alfo 
that of the prolate cycloid, is obtained from the above. 
For put a = the circumference of the circle v p r, b = 
the bafe d g, 3 = the arch v p, s = p m, and y ±= f m • 

and then a : b :: z : — = pf. Confequentlyjr = — 

+ 5, the equation of the curtate and alfo that of the 
prolate cycloid. 




SECTION 









t 257 ] 


SECTION V. 

Of the Logarithmic Curve . 


DEFINITIONS. 

I. 

If in the ftraight line x y, of an unlimited length. Fig. 
fegments ab,bc,cd 8cc. be taken equal to one an¬ 
other, but indefinitely fmall, and if from the points of 
fe&ion perpendiculars a e, b f, c g, d h Sec. be drawn, 
and be in geometrical progreffion; the perpendiculars 
will be indefinitely near to one another, and the line 
drawn through their extremities e, f, g, h, Sec. is 
called the Logarithmic Curve. 

II. 

The ftraight line x y is called the Axis to the loga¬ 
rithmic curve, and the perpendiculars ae, b f, c g, 

D H, &c. are called Ordinates to it. 

PROPOSITION. 

The axis is an afymptote to the logarithmic curve . 

For, the reft remaining as in the Definitions, let the Fi s- 
ordinates b f, c g, d h, Sec. on the right of a e conti¬ 
nually increal'e; and, the fegments Ah, be, cd, Sec. in. 
the axis being equal to one another, and each equal to 
A b, let the ordinates hf eg, dh,Si c. on the left of 
A e continually decreafe 5 that is, let b f be to a e as 
a e to bf, and let a e, hf, eg, dh, Sec. be in geo¬ 
metrical progreffion. Then ending the firft rank, and 
beginning the fecond with ordinates equally diftant 
from a e, we have the two following ranks of magni¬ 
tudes proportional taken two and two in the fame order* 
s a E 


r 



OF THE LOGARITHMIC CURVE, 


*58 

SECT. 

V. 


ae:bf:cg:dh 
dh :c g : bf : A e, 

and therefore (22. v.) a e : d h : : db : A e ; or d h j 
a E : : a e : d k. Hence it is evident that the re&- 
angle under any two ordinates equally diflant from A e 
is equal to the fquare of a e ; and therefore if an ordi¬ 
nate on the right of a e be indefinitely great, an ordi¬ 
nate on the left of A e, and equally diflant from it, will 
be indefinitely fmall, but it can never become equal to 
nothing, or vanifh. Confequently the axis x y is an 
afymptote. 

SCHOLIUM, 

As A b, a c, A d, &c. conflitute a feries in arithme¬ 
tical progreflion, and a e, b f, c g, d h, Sec. a corre- 
fponding feries in geometrical progreflion, thefegments 
A b, A c, a d, &c. are analogous to a feries of natural 
numbers, and the ordinates b f, c g, d h, &c. are ana¬ 
logous to the logarithms of thefe numbers. The curve 
is named from thefe analogies. 

The equation of the curve is deduced from the firft 
Definition, in the following manner. Put a e = 1, 
B f = a, and then 1 : a : : a ; a 1 = c g ; and a : a 2 : : 
a 1 : a 3 = d h, &c. Hence it is evident that If x de¬ 
note any number of equal parts ab,bc,cd in the 
axis, then will a x be equal to the ordinate drawn 
through the extremity of the fegment in the axis de¬ 
noted by x) and therefore if this ordinate be put equal 
to y> then a* =j', which is the equation of the curve. 

The logarithmic curve, on account of its equation, 
is alfo called an Exponential Curve • 


OP 



[ 2 59 3 


OF SPIRALS, 


SECTION VL 

Of the "Logarithmic Spiral. 


DEFINITIONS. 

I. 

If any number of ftraight lines ca, cb, CD, ce, See. Fig. 23, 
drawn from the point c within the curve a b d e con¬ 
tain equal angles with it, the curve is called the Loga¬ 
rithmic Spiral * 

it. 

The point c is called the Center of the Logarithmic 
lpiral; and any ftraight line drawn from the center to 
the curve is called an Ordinate. 

PROPOSITION* 

Any number of ordinates of the logarithmic curve are iii 
geometrical progrejjion , if they contain equal angles at 
the center. 

Let abde be a logarithmic curve* of which c is Fig. 13. 
the center and ca,cb,cd,ce, 8tc. ordinates, and let 
the angles acb, bcd, dc e, See. be equal to one 
another; the ordinates ca, cb, cd, ce, See* are in 
geometrical progreflion. 

For let the angles acb, b c d, d c e, 8ec. be indefi¬ 
nitely fmall, and then the portions ab, b d, d e, &c. 
of the curve may be confidered as ftraight lines, and as, 

S3 by 




z6o 

SECT. 

VI. 


OF THE LOGARITHMIC SPIRAL. 

by the firft Definition, the angles cab,cbd 3 cde, 
See. are equal, the triangles acb, bcd, dce, See. are 
equiangular. Confequently (4. vi.) A c : b c : : b c : 
d c; and bc:dc::dc:ec. The Propofition is 
therefore evident. 

Cor . 1. If the ordinates b c, d c, e c, See. on the 
right of a c fucceffively increafe, the ordinates b c, d c, 
ec , See. on the left of a c will fucceffively decreafe. 
For the reft remaining as above, if the angles acb, 
a c b , b c d, d c e , be equal to one another, it may be 
proved as above that b c : a c : : a c : b c, and a c : 
be: : b c : d c, See. 

Cor. 2. It may be proved, as in the logarithmic 
curve, that a c is a mean proportional between ordi¬ 
nates equally diftant from it; and therefore as the lo¬ 
garithmic curve cannot meet its afymptote at any al- 
figned diftance, fo the logarithmic fpiral cannot fall in¬ 
to its center at any affigned number of revolutions. 

SCHOLIUM. 

As the angles ace, a c d, a c e, See. conftitute a 
feries in arithmetical progreffion, and the ordinates ac, 
b c, d c, e c, &c. a feries in geometrical progreffion, 
the ordinates are analogous to natural numbers, and the 
angles to their logarithms; and from this confideration 
the fpiral receives its name. 

If a c be put = 1, b c = a } x = any angle in the 
arithmetical feries, and^y = the correfponding ordinate, 
then a x = y, the equation of the logarithmic fpiral, for 
the fame reafons as ftated in the fcholium on the loga¬ 
rithmic curve. 


SECTION 



r 261 ] 


SECTION VII. 

Of the Spiral of Archimedes, 


DEFINITIONS. 

r. 

If the ftraight line c l revolve in a plane about one fig* *5* 
of its extremities c as a center, faith an uniform velo¬ 
city, and if at the commencement of the revolution a 
point begin to move from c and proceed in cl to¬ 
wards l with an uniform velocity, the line defcribed 
by the point moving in c l is called the Spiral of Ar¬ 
chimedes. 

II. 

The line c g a defcribed in the firft revolution of 
c L is called the Firft Spiral ; the line A h b defcribed 
in the fecond revolution of c l is called the Second Spi¬ 
ral ; the line b k d defcribed in the third revolution 
of c l is called the Third Spiral , &c. 

III. 

If a be a fixed point in c l, and c A be the diftance 
moved over by the deferibing point during the firft re¬ 
volution, the circle defcribed by the point a during 
the revolution is called the Generating Circle ; and 
the point c is called the Center ol any one of the lpi- 
rals. 

IV. 

A ftraight line drawn from c to the point in which 
any one of the fpirals ends is called the Axis of that 
fpiral; and a ftraight line drawn from c to any other 
pcint in the fpiral is called an Ordinate . 

s 3 


PROP. 




z6z 


OP THE ARCHIMEDEAN SPIRAL 


SECT. 

VII. 


PROP. I. 

If an ordinate be drawn to any point in thefrjl fpiral^ 
and be produced till it cut the generating circle , the or¬ 
dinate will be a fourth proportional to the circumference 
of the generating circle , the axis , and the circular arch 
generated from the beginning of the revolution to the 
point of faction* 


Fig. 16. Let A g be an ordinate to the firft fpiral a g b, of 
which b K h z is the generating circle, and a being 
the center, let a b be the axis, and let ag be produced 
and cut the circle in H } the ordinate ag is a fourth 
proportional to the circumference b k h z, the axis 
a b, and the circular arch b k h, generated from the 
beginning of the revolution to h the point of fe&ion. 

For the line in which the fixed point b is fituated 
revolves about a with an uniform velocity, and the 
point deferibing the fpiral moves from a towards B 
with an uniform velocity, according to the third and 
firfl Definitions. The circumference of the circle 
therefore and the axis a b are deferibed by uniform 
velocities in the fame time; and for the fame reafons 
the arch b k h and the ordinate a g are deferibed by 
the fame uniform velocities in the fame time. But 
fpaces deferibed by the fame uniform velocity are to 
one another as the times in which they are deferibed, 
and therefore the circumference of the circle b k h z 
is to the arch b k h as the time of one revolution to 
the time of deferibing the arch b k h, and a b is to 
a g as the fame portions of time are to one another. 
Confequently (n. v. and alternation) the circumfe- t 
rence of the circle b k h z is to the axis ab as the arch 
B K h to the ordinate a g. 

Cor. i. The reft remaining, if a f another ordinate 
be produced and cut the generating circle in z, by the 

above 



OP THE ARCHIMEDEAN SPIRAL. 


263 

above (and 11 . v.) the arch b k h : the arch b k z : : SECT. 
A g : a f. vu * 

Cor . 2. If the axis of the firft fpiral, or, which is the 
fame thing, the radius of the generating circle, be put 
equal to r, the circumference of the generating circle 
equal to c, any arch b k h from the axis equal to z, 
and the correfponding ordinate a g equal to y y then 

c ; r : 1 z : ~ —y, the equation of the firft fpiral. 

PROP. II. 

An ordinate drawn to any point in the fecond fpiral is a 
fourth proportional to the circumference of the generating 
circle , the radius of the circle , and the fum of the cir¬ 
cumference of the circle and its arch generated from the 
beginning of the revolution to the ordinate . 

Let a e be an ordinate drawn to any point e in the Fig. z6< 
fecond fpiral bed m, and let it cut the generating 
circle b k h z in h. Let a be the center and a m the 
axis, and confequently the fituation from which the 
revolution begins. The circumference b k h z of the 
generating circle is to its radius a b as the fum of the 
circumference b k h z and the arch bkh to the ordi¬ 
nate A E, 

For, as a l revolves with an uniform velocity, the 
circumference b k h z is to the fum of the circumfe¬ 
rence b K h z and the arch bkh as the time of one 
revolution to the time in which the circumference and 
the arch bkh are generated by b the extremity of 
the radius. And thefe portions of time are to one an¬ 
other as the radius a b to the ordinate a e, as the point 
moving in a l with an uniform velocity partes over a b, 
a e in thefe portions of time refpe&ively. Confequently 
(11. v. and alternation) the circumference b k h z : 

the 





OF THE ARCHIMEDEAN SPIRAL. 


SECT. 

VII. 


the radius ab : : the circumference bkhz| the arch 
B k h : the ordinate a e. 

Cor . i. The reft remaining as above, if a d another 
ordinate to the fpiral b e d m cut the generating circle 
in z, then, by the above, (and n. v.) the circumfe¬ 
rence b k h z + the arch b k h : a e : : the circum¬ 
ference b k h z + the arch b k z : a d. 

Cor. 2. Put c = the circumference of the generating 
circle, r = its radius, y = the ordinate a e, and z = 
the arch b k h ; and then, by the Propofition, c -h z i 
y : : c : r, and therefore the equation of the fecond fpi¬ 
ral is r X c + z — cy. 

Cor. 3. If c denote the circumference of the gene¬ 
rating circle, r its radius,^; an ordinate drawn to any 
point in the nth fpiral, and z the arch of the generat¬ 
ing circle generated from the beginning of the revolu¬ 
tion to the ordinate, it may be proved, as above, 
that c : r : : n — 1 x c + z : y. Confequently r x 
n c — c + z = cy is a general equation for any one 
of the fpirals of Archimedes. 

Cor. 4. The reft of the notation remaining as above, 
let an ordinate v contain an angle with the ordinate y > 
and let d denote the arch in the generating circle 
which meafures this angle. Let z denote an arch of 
the generating circle, generated from the beginning of 
the revolution to the ordinate y ; and let the ordinate 
Y and the ordinate x contain an angle equal to the an¬ 
gle contained by y and v, and confequently alfo mea- 
fured by an arch d in the generating circle. Then by 
the laft Cor. (and 11. v.) n — 1 x c + z :,y : : n — 1 x 
c + z 4 - d ': v ; : n —1 xr+z : y: : rc — 1 xc+z 
+ 4 : V 5 and therefore, by Lerpma X. page 154. (and 
ji. v.) d : v —y : : d ; x — y. Confequently (14. v.) 

the 







OP THE ARCHIMEDEAN SPIRAL. 


the excefs of v above y is equal to the excefs of x 
above y. 

Cor. 5. From the laft Cor. it is evident, that if any 
number of ordinates contain angles which conftitute a 
feries in arithmetical progreffion, the ordinates them- 
felves will be in arithmetical progreffion. 



Pit 


r: tji n 



SECTION 



SECTION VIII. 

-Of the Hyperbolic or Reciprocal Spiral, 


DEFINITIONS. 

I. 

If with c, one of the extremities of the ftraight line 
c l, as a center and diftances ca,cb,cd, &c. in cl, 
arches of circles A g, b h, d i, &c. be defcribed, and 
if thefe arches be equal to one another, and indefinite¬ 
ly near; the curve g h i, he. drawn through their ex¬ 
tremities is called the Hyperbolic or Reciprocal Spiral . 

II. 

The ftraight line c l is called the Axis of the hyper¬ 
bolic fpiral, the point c its Center , and any ftraight 
line c k drawn from the center to the curve is called 
an Ordinate, 


PROP. I. 

In the hyperbolic fpiral any tivo ordinates are reciprocally 
to one another as the angles-they co?itain with the axis. 

Let e g i be a hyperbolic fpiral, of which c is the 
center, c l the axis, and c i, c g any two ordinates $ 
c i is to c g as the angle l c g to the angle pci. 

For let the circular arches g a, i d be defcribed, 
and let them meet the axis in a and d. Let c g pro¬ 
duced meet the arch d i produced in m ; and then as 
D c m, a c g are fimilar fe&ors, by Cor. 3. to the 
Lemma in the third fe&ion, c d : c A : : the arch d m : 
the arch a g. But c 1 is equal to c d, and c GtocA, 



OP THE HYPERBOLIC SPIRAL. 


and by the firft Definition the arch a g is equal to the SECT, 
arch d i; and therefore c i : c g : : the arch d m : the VHL 
arch d i. Again (33. vi.) the arch d m : the arch d i 
: : the angle DCMortco: the angle DCiorLCi, 
Confequently (11. v.) c 1 is to c g as the angle l c g 
to the angle lci. 

Cor. Every thing remaining as above, let c 1 be a 
given ordinate, and put it equal to r; put the arch d i 
= fl,co=;, and the arch d m = %. Then, by the 
above, r :y : : z : a, and ar —yz, the equation of the 
curve. 


PROP. II. 

If from the center of an hyperbolic fpiral a Jlraight line be 
drawn at right angles to the axis and equal to one of 
the circular arches intercepted between the curve and 
axis , a Jlraight line drawn through the extremity of this 
perpendicular parallel to the axis will be an ajymptote 
to the curve . 

Let e g 1 be an hyperbolic fpiral, of which c is the 2 ^ 
center and c l the axis, and let the ftraight line c b at 
right angles to c l be equal to 1 d any circular arch 
intercepted between the curve and the axis; the 
ftraight line b f drawn through b parallel to c l is an 
afymptote to the curve. 

For through 1 draw n f parallel to c b, or perpendi¬ 
cular to c l, and let it meet b f in f ; and then n v 
{34. i.) is equal to c b, and therefore equal to the arch 
d 1. But the arch d i is greater than its fine n i, and 
therefore the point f in b f is without the curve. 

Again, from the preceding Propofition it is manifeft, 
that if the ordinate c 1 be indefinitely great when com¬ 
pared to another ordinate c g, the angle d c 1 is inde¬ 
finitely fmail when compared to a c g ; and when an 

angle 



26S 


o£ THE HYPERBOLIC SPIRAL. 


SECT. 

VIII. 


angle is indefinitely fmall, the excels of the arch which 
meafures it above its fine is lefs than any affigned mag¬ 
nitude. Cortfequently as i f is the excefs of the arch 
d i above its fine N i, if the curve e g i be continued 
till the angle dci become indefinitely final!, the dif- 
tance i f of the curve from b f will be lefs than any 
affigned magnitude. The ftraight line b f is therefore 
an afymptote to the curve. 


the: END. 






Plate \\\III/xu/t *68 . 


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